Frege's Theory of Real Numbers: A consistent Rendering
aa r X i v : . [ m a t h . L O ] J a n Frege’s Theory of Real Numbers: A Consistent Rendering
FRANCESCA BOCCUNI ∗ & MARCO PANZA † Abstract
Frege’s definition of the real numbers, as envisaged in the second volume of
Grundgesetzeder Arithmetik , is fatally flawed by the inconsistency of Frege’s ill-fated
Basic Law V . Werestate Frege’s definition in a consistent logical framework and investigate whether it canprovide a logical foundation of real analysis. Our conclusion will deem it doubtful that sucha foundation along the lines of Frege’s own indications is possible at all.
The aim of the present paper is twofold: ( i ) rephrasing Frege’s inconsistent definition of realnumbers, as envisaged in Part III of Grundgesetze der Arithmetik (Frege 1893-1903), in a con-sistent setting ruling out value-ranges, and so involving no version of the infamous
Basic Law V (BLV); ( ii ) wondering whether the rephrased definition can be considered logical, and, as such,as a ground for a logicist view about real analysis.Concerning ( ii ) a proviso is in order. In the debate on neologicism, a distinction has beenmade between logicality and analyticity, by suggesting, for instance, that, though not logical,Hume’s Principle (HP) is analytic. We are far from undermining the relevance of this distinction,but we consider unnecessary to stress it for our present purpose. There are two reasons for that.On the one side, we deem all the arguments we will advance against the logicality of the relevantprinciples and definitions also apt to oppose their analyticity—though some of those advancedin favor of the former are possibly only sufficient to argue for the latter. On the other side, weare interested in the epistemic attitude that a faithful Fregean (or even Frege himself) mighthave (had) in the face of a definition such as our own. Hence, for the sake of our discussion, wemust follow Frege himself in taking a “truth” to be analytic if, in its proof, “one only runs intological laws and definitions” (Frege 1884, §
3; Frege 1953, p. 4) , and in regarding definitions asmandatorily explicit, which suggests regarding logicality as a necessary condition for analyticity,rather than the latter as a weaker condition than the former.Concerning ( i ), it is important to observe that, for Frege, real numbers had to be defined asratios of magnitudes, and magnitudes had to belong to different domains. Hence, his definitionshould have included two successive steps: a definition of domains of magnitudes, and a definitionof ratios on these domains. As a matter of fact, he accomplished only the former step, and merelygave some informal indications on how to accomplish the latter. Both things are done in the ∗ Vita-Salute San Raffaele University, Milan. Orcid: 0000-0001-9814-1431. † IHPST (CNRS and Universit´e de Paris 1, Panth´eone Sorbonne), and Chapman University, Orange, CA. In what follows we will use ‘definition’ quite broadly. The meaning of its occurrences will be clarified in context. We slightly modify Austin’s translation.
Grundgesetze . The latter step should have presumably been accomplished ina third volume that, once aware of Russell’s paradox, Frege never wrote.Had he accomplished this step, he should have made it conform to a crucial requirement:supposing that several domains of magnitude exist, the definition of real numbers should haveidentified a ratio on one of these domains with the same real number as a ratio on each one ofthe others. Our definition actually complies with this requirement.After a short presentation of Frege’s strategy in §
2, we will consistently rephrase his definitionof domains of magnitudes in § . To eliminate value-ranges, we will rephrase first-order formulasinvolving terms for them as higher-order formulas proper to a system of higher-order predicatelogic as weak as possible, on which we will take stock in §
4. This seems to us the most faithfulway to consistently render Frege’s original definition. Insofar as our appreciation of the logicalityof our definition depends on assuming, in a genuinely Fregean vein, the logicality of higher-orderlogic, we contend that this appreciation ipso facto provides an appreciation of the logicality ofFrege’s own definition that remains perfectly independent of any judgement about the logicalityof (any consistent version of) BLV .In §
5, we will investigate how to define real numbers by following Frege’s indications, on thebase of our definition of domains of magnitudes. In § II.164 of his treatise, Frege explicitly ac-knowledged that his envisaged definition of these numbers requires an existence proof of nonemptysuch domains. We will explain why this is so. Here, it is only in order to observe that, in thissame § , he also argues that this existence depends on the existence of continuously many objects(an infinity of objects larger than “ Endloss ”, the cardinality of “finite cardinal numbers”), andsketches a plan for this proof, which, taking the existence of natural numbers for granted, aims atconstructing these objects from them. He then claims that, thanks to this proof, he would havesucceeded “in defining the real number purely arithmetically or logically as a ratio of magnitudesthat are demonstrably there” (Frege 2013, p. 162 ).The adverb “arithmetically” is clearly used to emphasize that the envisaged definition wouldhave been independent of both empirical considerations and geometry. In this sense, the defini-tion would have surely been arithmetical, and our rendering of it will be as well. But there isanother sense in which, despite his appealing to natural numbers, Frege did not certainly want hisdefinition to be arithmetical: both his criticisms to the alternative definitions depending on anextension of the domain of rationals—including Cantor’s ( §§ II.68-85), Dedekind’s ( §§ II.138-147),and Weierstrass’s ( §§ II.148-155)—and the very purpose of identifying real numbers with ratiosof magnitudes make clear he wanted these numbers to be strictly independent of natural ones, tobe properly
Zahlen , rather then
Anzahlen . By offering our definition, we will try, among otherthings, to comply with this requirement.In §
6, we will account for two distinct strategies to get the required existence proof in oursetting. One of them conforms to Frege’s indications, while the other might be considered moreappropriate for ensuring logicality, since, pace
Frege, it does not require that the existence ofcontinuously many objects be established. In §
7, we will investigate whether the resulting Frege (1893-1903), §§ II.165-II.245 and § II.164, respectively; §§ II.55-II.159 contain a critical discussion ofalternative definitions, while §§ II.160-II.163 contain an informal introduction and a principled justification of thedefinition of domains of magnitudes. Knowledge of Frege’s original definition is required to appreciate its correspondence with our rephrasing. Usefulaccounts of it can be found in Dummett (1991, ch. 22); Schirn (2013); Simons (1987); Shapiro & Snyder (2020). Anyone supporting Quine’s view on the non-logicality of higher-order logic can take our granting it as madefor the sake of the argument. §
8, wewill provide some concluding remarks.
Frege’s strategy agrees with the “application constraint”: the requirement that a mathematicaltheory be shaped as to immediately account for its applications . This motivates his suggestionto define real numbers as ratios of magnitudes, magnitudes as elements of distinct domains sup-posedly including those of geometric, mechanic and empirical ones, and ratios on these domainsas measures of the relevant magnitudes. Insofar as it would be odd to require that the theoryof real numbers involve these magnitudes as such, together with their respective theories, thismakes providing a structural definition of domains of magnitudes mandatory: a definition thatmerely fixes the conditions that a certain domain of independent items has to meet in order tobe recognized as a domain of magnitudes. Frege himself clearly stresses this crucial point (Frege1893-1903, § II.161; Frege 2013, p. 158 ):There are many different kinds of magnitudes: lengths, angles, periods of time,masses, temperatures, etc., and it will scarcely be possible to say how objects thatbelong to these kinds of magnitudes differ from other objects that do not belong to anykind of magnitude. Moreover, little would be gained thereby; for we still lack any wayof recognizing which of these magnitudes belong to the same domain of magnitudes.Instead of asking which properties an object must have in order to be a magnitude,one needs to ask: how must a concept be constituted in order for its extension to bea domain of magnitudes?A natural way to render the required structural definition would have provided definitionalaxioms, as usually done for groups or fields. An informal conception of magnitudes recognizing theexistence of “lengths, angles, periods of time, masses, temperatures, etc.” might have suggestedthat there are non-isomorphic models satisfying these axioms. Still, for Frege, magnitudes are justthose items that real numbers are ratios of, and they all behave as lengths do, so that domains ofmagnitudes are all isomorphic to each other. Had he defined them through appropriate axioms,these should have then been expected to be categorical, though algebraic in nature—as it happensfor the usual axioms for real numbers themselves, namely the axioms of a totally ordered andDedekind-complete field. Moreover, insofar as magnitudes are required to add to each other butnot to multiply with each other (namely to admit only a single internal composition law), whathe would have needed is a categorical axiomatization for totally ordered, dense and Dedekind-complete (and, then, also Abelian and Archimedean) groups.Frege did not straightforwardly follow this route, however. Conforming with a remark byGauss (1931, p. 635, also in Werke , II, pp. 175-76; quoted in Frege, 1893-1903, § II.161) andputting it in his perspective, he conceives of magnitudes as value-ranges of permutations, and sodefines their domains not as domains of items merely satisfying certain conditions, but rather asdomains of extensions of appropriate first-level binary relations satisfying these conditions. Thismakes him able to appeal, along with his definitions, to structural properties of first-level binary See Panza & Sereni (2020) and Sereni (2019), which include a critical survey of the recent discussion on Frege’sattitude toward applications of mathematical theories. See footnotes 19 and 28 below. ξ ⌢ ζ , often too quickly identified with set-theoretic membership, whose definition islicensed by BLV. Once BLV is omitted, this function can no more be defined, and the reduction tofirst-order is no more possible—unless by a form of set theory. Hence, making Frege’s definitionconsistent by eliminating BLV without falling into a set-theoretical setting requires replacingFrege’s first-order definitions with higher-order ones. We will explain how this can be done byappropriately rephrasing Frege’s definitions, and in clarifying the logical nature of the (logical)system that is required for that. This is the purpose of the next two §§ . The omission of BLV is made possible by the elimination, from Frege’s language, of terms forvalue-ranges. Insofar as the presence of these terms in his definition of domains of magnitudesentirely depends on the function ξ ⌢ ζ , we have to make its use pointless.To make a long story short, this function is such that for any objects Γ and ∆, if Γ is thevalue-range , ε Φ ( ε ) of a first-level one-argument function Φ ( ξ ) , then ∆ ⌢ Γ is Φ (∆), and if Γ isnot such a value-range—or, better, it is not a value-range at all, since, in Frege’s formalism, anyvalue-range reduces to the value-range of a first-level one-argument function—, then ∆ ⌢ Γ isthe value-range of a first-level concept under which no object falls—for example that of ¬ ( ξ = ξ ),which we could denote by ‘ ⊘ ’, for short. In other terms, ∆ ⌢ Γ is the value, for ∆ as argument,of the first-level one-argument function of which Γ is the value-range, if Γ is a value-range, and ⊘ , if it isn’t—whatever object ∆ might be.In a rich enough second-order predicate language including the operator ‘ ιz [ z : ϕ ]’ for definitedescriptions, together with a symbol for value-ranges, the individual variables ‘ x ’, ‘ y ’ and ‘ z ’, andthe monadic predicate one ‘ F ’, this stipulation could be rendered as follows: ∀ x, y (cid:2) x ⌢ y = ιz (cid:2) z : ∃ F (cid:0) y = , εF ( ε ) ∧ F ( x ) = z (cid:1)(cid:3)(cid:3) , provided that ‘ ιz [ z : ϕ ]’ designates a well-defined object, namely ⊘ , even if there is no z suchthat ϕ . If ‘ a ’ and ‘ b ’ are terms, this makes ‘ a ⌢ b ’ be a term in turn.This licenses using this term to denote the ζ -argument of the same function ξ ⌢ ζ . Taking anew term ‘ c ’ to denote the ξ -argument, one has the new term ‘ c ⌢ ( a ⌢ b )’ such that c ⌢ ( a ⌢ b ) = ιz (cid:2) ∃ G (cid:0) b = , α , εG ( ε, α ) ∧ G ( c, a ) = z (cid:1)(cid:3) . Our use of Greek capital letters to denote objects and functions whatsoever corresponds to Frege’s, in his“exposition” of his formal language (Frege 1893-1903, Part I, § I.1-52).
4t follows that ‘ c ⌢ ( a ⌢ b )’ is a term that denotes the value for c and a as arguments of thefirst-level two-argument function of which b is the value-range, if b is such a value-range .Hence, if Φ ( ξ ) and Ψ ( ξ, ζ ) are a first-level one-argument and a first-level two-argument func-tion, respectively, then a ⌢ , ε Φ ( ε ) = Φ ( a ) c ⌢ (cid:0) a ⌢ , α , ε Ψ ( ε, α ) (cid:1) = Ψ ( c, a ) . Suppose that ‘ P b ’ and ‘ R b ’ be respectively a monadic and a dyadic predicate appropriatefor rendering, in an appropriate predicate language, two functions Φ ( ξ ) and Ψ ( ξ, ζ ) of which b is the value-range. It follows that, in order to make the use of the function ξ ⌢ ζ pointless,and so eliminate value-ranges while restating Frege’s definition of domains of magnitudes, it isenough to replace each term of the form ‘ a ⌢ b ’ with the formula ‘ P b a ’ and each term of the form‘ c ⌢ ( a ⌢ b )’ with the formula ‘ c R b a ’, and to transform Frege’s formal system accordingly .The system so obtained will be independent of BLV, and so will any definition stated in it. Informally speaking, Frege conceived of a nonempty domain of magnitudes as a totally ordered,dense and Dedekind-complete additive group of permutations. In light of his rejection of implicitdefinitions, defining such a group required to explicitly defining a particular function to playthe role of its (additive) law of composition, which required, in turn, to have objects available,endowed with an internal structure making such a definition possible. To this purpose, he madethe simplest choice possible: he took those objects to be extensions of functional first-level binaryrelations, and assigned the role of this law to the composition of the corresponding relations. Thisobviously resulted in taking the extension of the identity relation as the neutral element of thegroup, and the extensions of the inverse relations as its inverse elements.This choice is easily rendered in a predicate setting lacking extensions. We merely have to fixthe conditions under which a first-level binary relation is functional and results either from theinversion of another such relation, or from the composition of two other such relations. Taking ‘ R ’and ‘ S ’ to range over first-level binary relations, this can be formally done through the followingexplicit definitions:[Functionality] ∀ R ( I R ⇔ ∀ x, y ( xRy ⇒ ∀ z ( xRz ⇒ y = z ))) , [Inversion] ∀ R ∀ x, y ( xR − y ⇔ yRx ) , [Composition] ∀ R, S ∀ x, y ( x [ R ⊔ S ] y ⇔ ∃ z ( xRz ∧ zSy )) . These definitions define three third-order constants, respectively: the monadic predicate con-stant ‘ I ’, designating a property of first-level binary relations; the monadic functional constant If b is not such a value-range, different cases are possible. It is not necessary to account for them, here. For acomplete treatment, see Panza (FC1). Here and in what follows, we take boldface capital latin letters as dummy letters for first-level properties andrelations. The same letters in italics will, instead, be used for the corresponding variables. To be sure, this rendering of the function ξ ⌢ ζ in terms of predication is not fully faithful to Frege’s originalview: for Frege, both ‘∆ ⌢ Γ’ and ‘—∆ ⌢ Γ’ (see footnote (12), below) denote an object, while for us ‘ a ⌢ b ’is rather a formula. Nevertheless, whenever ‘Γ’ denotes the extension of a concept Φ ( ξ ), ‘∆ ⌢ Γ’ and ‘—∆ ⌢ Γ’are, for Frege, names of the True if and only if ‘Φ(∆)’ itself is a name of the True. Insofar as only this case isrelevant here, this rendering does not alter the aspects of Frege’s definition that are of interest here. ∼ − ’, designating a one-argument function from and to first-level binary relations; the dyadicfunctional constant ‘ ∼ ⊔ ∼ ’, designating a two-argument function from and to first-level binaryrelations, again. Hence, in order to be licensed, they require appropriate instances of predicativecomprehension. The first requires the following instance of third-order predicative comprehensionwithout parameters:(Functionality-CA) ∃ X ∀ R ( X R ⇔ ∀ x, y ( xRy ⇒ ∀ z ( xRz ⇒ y = z ))) . The other two require the following instances of second-order dyadic predicative comprehensionwith parameters respectively:(Inversion-CA) ∀ R ∃ S ∀ x, y ( xSy ⇔ yRx ) , (Composition-CA) ∀ R, S ∃ T ∀ x, y ( xT y ⇔ ∃ z ( xRz ∧ zSy ))) , where ‘ T ’ ranges over first-level binary relations, too.One might replace, however, these explicit definitions with mere (metalinguistic) typographicstipulations: (Functionality ′ ) I ( R ) := ∀ x, y ( xRy ⇒ ∀ z ( xRz ⇒ y = z )) , (Inversion ′ ) R − ( xy ) := yRx, (Composition ′ ) R ⊔ S ( xy ) := ∃ z ( xRz ∧ Szy ) . Any instance of the left-hand side of these stipulations is intended to be a mere abbreviation of thecorresponding instance of the right-hand side. For example, while ‘ I R ’ in (Functionality) is anatomic third-order (open) formula, ‘ I ( R )’ in (Functionality ′ ) is an atomic symbol abbreviatingthe second-order (open) formula ‘ ∀ x, y ( xRy ⇒ ∀ z ( xRz ⇒ y = z ))’ . And analogously for ‘ xR − y ’and ‘ R − ( xy )’ in (Inversion) and (Inversion ′ ), respectively, and for ‘ x [ R ⊔ S ] y ’ and ‘ R ⊔ S ( xy )’in (Composition) and (Composition ′ ), respectively. Adopting these stipulations requires neitherany instance of comprehension, nor any extension of the usual second-order language.We will see in what follows whether these stipulations are enough for our purpose, or thecorresponding explicit definitions are needed, and the instances of comprehension they require . In Frege’s original setting things would not be so simple. Consider only the example of (Functionality). Inthis setting, the role of this definition is played by the definition of the first-level concept I ξ (Frege, 1893-1903, § I.37). By adapting Frege’s notation to our modern one, the definition might be stated as follows: (cid:2) ∀ x, y (cid:2) — ( x ⌢ ( y ⌢ a )) ⇒ ∀ z (cid:2) — ( x ⌢ ( z ⌢ a )) ⇒ y = z (cid:3)(cid:3)(cid:3) = I a. where ‘ a ’ is a term used as a parameter, and — ξ is the horizontal concept ( ibidem , § I.8), which is suchthat —Γ is the True if Γ is also the True, and the False otherwise. It follows that I a is the same object as ∀ x, y (cid:2) — ( x ⌢ ( y ⌢ a )) ⇒ ∀ z (cid:2) — ( x ⌢ ( z ⌢ a )) ⇒ y = z (cid:3)(cid:3) , which is a truth-value. If a is not a value-range of afirst-level binary relation, — ( b ⌢ ( c ⌢ a )) is the False for whatever pair of objects b and c , and I a is then theTrue, which makes any object other than a value-range of a first-level binary relation fall under the concept I ξ . If a is the value-range of a first-level binary relation Φ ( ξ, ζ ), a falls under the concept I ξ if and only if either Φ ( ξ, ζ )is empty, or, for any x , there is at most one y such that Φ ( x, y ) is the True. Clearly, there is no way to regard thisdefinition as a mere typographic stipulation. It rather defines a total first-level concept by introducing a functionalconstant to designate it. Among many others, there are two relevant differences with our case: i ) Frege’s definitionapplies in general, whereas both (Functionality) and (Functionality ′ ) only apply to first-order binary relations; ii ) Differently from (Functionality ′ ), Frege’s definition is licensed only via a stipulation analogous to second-ordercomprehension. Mutatis mutandis , this also applies to (Inversion) and (Composition), and to any other particulardefinition entering his definition of domains of magnitudes. .3 Domains of Classes For Frege, a domain of magnitudes is the domain of a “positive class”, which is in turn a “positivalclass” of an appropriate sort. In his setting, a class is the extension of a first-level concept(Frege, 1893-1903, § II.16), and the objects falling under this concept are said to belong to theclass. Positival and positive classes are, in particular, extensions of concepts under which (only)extensions of first-level binary relations fall. Defining them amounts to fixing the conditions thata concept is to meet for the objects falling under it to be just these extensions. To do this, Fregeappeals to their “domains”. He has, then, to firstly define, in general, domains of classes ( ibidem , § II.173). The definition applies to any class, but we only need to consider its application to thecase of the domain of a class of extensions of first-level binary relations.This is the extension of a concept under which fall: the extensions in the class; the extensionsof the inverses of the relations whose extensions are in the class; and the extensions of the relationscomposed by each of the relations whose extensions are in the class and their inverses—which incase these relations are functional, as required for both positival and positive classes, all coincidewith the extension of the identity relation. In our setting, we can, then, rephrase, Frege’s definitionof the domain of a class of extensions of first-level binary relations as follows:(3.1) ∀ X ∀ R ð X R ⇔ X R ∨∃ S (cid:20) X S ∧ ∀ x, y (cid:20) [ xRy ⇔ S − ( xy )] ∨ [ xRy ⇔ S ⊔ S − ( xy )] (cid:21)(cid:21) where ‘ X ’ is a third-order monadic variable, and ‘ ð ’ a functional operator applied to it. Thisdefinition makes clear that, when applied to whatever (second-level) property Q of first-levelbinary relations , ð gives another property ð Q of these same relations.To license this definition, we need to ensure the existence and uniqueness of a second-levelproperty providing a putative value for ð X under the existence of a second-level property pro-viding a value for X , and this requires, in turn, third-order comprehension with parameters. Butsuppose we wanted to define a certain (third-level) property Q that a class of first-level binaryrelations should have in order to be positival, which is required to render Frege’s definition ofpositival classes. If, in defining it, we had to appeal to the domains of the classes that could haveit, as is also required to render Frege’s definition, we should have recourse to a definition like this: ∀ X [ Q X ⇔ φ ( ð X )]where ‘ φ ( ð X )’ stands for an appropriate formula involving the predicate ‘ ð X ’. Hence, insofar as,in our rendering of Frege’s definition of positival and positive classes and domains of magnitudes,this predicate would only appear in instances of formulas of the form ‘ ð X R ’, we can replace (3.1)with the following abbreviation stipulation(3.1 ′ ) ð ( X )( R ) := X R ∨∃ S (cid:20) X S ∧ ∀ x, y (cid:20) [ xRy ⇔ S − ( xy )] ∨ [ xRy ⇔ S ⊔ S − ( xy )] (cid:21)(cid:21) , then use appropriate instances of ‘ ð ( X )( R )’ instead of the corresponding instances of ‘ ð X R ’. Asa matter of fact, this stipulation is all we need for our present purpose, and it must be supplied Here and in what follows we use ‘ Q ’ as a dummy letter for second-level properties. Later we will also use ‘ A ’,‘ E ’, ‘ H ’, ‘ L ’, ‘ M ’ and ‘ P ’ for the same purpose. Here we use ‘ Q ’ as a dummy letter for third-level properties.
7y no sort of comprehension, since it introduces no new predicate, but merely lets each instanceof its left-hand side be an abbreviation of the corresponding instance of the right-hand side. Forshort, read both ‘ ð XR ’ and ‘ ð ( X )( R )’ as ‘ R belongs to the domain of the class of first-orderbinary relations that have X ’. We can now consider Frege’s definition of positival classes. If we had to render it through a(nexplicit) definition, we should define a fourth-order monadic predicate constant designating athird-level property. This would require to quantify over second-level properties, and, then, toappeal to fourth-order comprehension. But, once again, we are not forced to do it. As above, wemight recur to an abbreviation stipulation by so avoiding any sort of comprehension.In agreement with Frege’s definition, the extension of a first-level binary relation R belongs toa positival class if (and only if): both R and its inverse are functional; the extension of R ⊔ R − ,i.e. the identity relation, does not belong to the class; and for any first-level binary relation S , ifits extension belongs to the class, then: the class of the objects that bear R to some other objectcoincides with the class of the objects to which some object bears S ; the extension of R ⊔ S belongs to the class; both the extension of R − ⊔ S and that of R ⊔ S − belong to the domain ofthe class. In our setting, this can be rendered either this way(3.2) ∀ X L X ⇔ ∀ R X R ⇒ ∀ S X S ⇒ ∀ x [ ∃ y ( xRy ) ⇔ ∃ z ( zSx )] ∧ X R ⊔ S ∧ ð X R ⊔ S − ∧ ð X R − ⊔ S ∧ I R ∧ I R − ∧ ¬ X R ⊔ R − , or this way:(3.2 ′ ) L ( X ) := ∀ R X R ⇒ ∀ S X S ⇒ ∀ x [ ∃ y ( xRy ) ⇔ ∃ z ( zSx )] ∧ X R ⊔ S ∧ ð ( X ) ( R ⊔ S − ) ∧ ð ( X ) ( R − ⊔ S ) ∧ I ( R ) ∧ I ( R − ) ∧ ¬ X R ⊔ R − , where both ‘ L X ’ and ‘ L ( X )’ are short for ‘ X is a positival class’ or, more precisely, ‘thefirst-level binary relations having X form a positival class’.In (3.2), ‘ L ’ is a fourth-order predicate constant and ‘ L X ’ an atomic (open) formula. Thisis, then, an explicit definition, which is to be licensed by an appropriate form of fourth-ordercomprehension. In (3.2 ′ ), ‘ L ( X )’ is, instead, an abbreviated (open) formula, and ‘ L ’ is merely asymbol occurring in it. Hence (3.2 ′ ) neither requires a fourth-order language nor is to be licensedby any form of fourth-order comprehension. This makes the relations whose extensions belong to the class permutations on a subjacent first-order domain. − ’ and ‘ ⊔ ’ do not merely occur as parts of the abbreviated formulas‘ R − ( xy )’ and ‘ R ⊔ S ( xy )’ introduced by (Inversion ′ ) and (Composition ′ ), but as functional signsallowing to construe the predicate variables ‘ R − ’ and ‘ R ⊔ S ’. Their use in (3.2 ′ ) is, then, tobe licensed by the explicit definitions (Inversion) and (Composition), which respectively require,in turn, (Inversion-CA) and (Composition-CA), or, more generally, the following second-orderpredicative comprehension axiom schema with parameters:(PCA ) ∀ R . . . T ∃ U ∀ x, y h xU y ⇔ φ ∆ ( R . . . T ) i , where ‘ U ’, ‘ R ’, ‘ S ’, and ‘ T ’ range over first-level binary relations, and ‘ φ ∆ ( R . . . T )’ stands forany second-order formula containing the parameters ‘ R ’, . . . ‘ T ’, but no higher-order quantifiers.Before going ahead with the definition of positive classes, some remarks are in order about theinformal import of the conditions characterizing a positival class. They apply, mutatis mutandis ,both to (3.2) and to (3.2 ′ ), but, for short and simplicity, we only make them about the latter.Let L be a second-level property. Requiring that ∀ R (cid:2) L R ⇒ (cid:0) I ( R ) ∧ I ( R − ) (cid:1)(cid:3) amounts to requiring that both a binary relation that has L and its inverse are functional. Ifthis condition obtains, requiring that ∀ R (cid:2) L R ⇒ ¬ L R ⊔ R − (cid:3) and that ∀ R, S ∀ x [( L R ∧ L S ) ⇒ ∃ y [( xRy ) ⇔ ∃ z ( zSx )]]respectively amount to requiring that the identity relation has not L , and that all the relationshaving L are permutations on a subjacent unspecified set. Hence, only permutations but theidentity one, have L . Thus, ⊔ is an associative law of composition without neutral element onthe relations having L . Again, if all the above conditions obtain, requiring that ∀ R, S [( L R ∧ L S ) ⇒ L R ⊔ S ]amounts to requiring that the family of permutations having L is closed under ⊔ . This makes:the inverse of any such permutation not have L —since, if it did, the identity permutation wouldalso have it; the family of permutations that satisfy the open formula ‘ ð ( L )( R )’ be also closedunder composition of the inverses of those having L —since, for whatever permutations R and S that have L , R − ⊔ S − is the same permutation as ( S ⊔ R ) − . All this is still not enough toensure that the family of permutations that satisfy the open formula ‘ ð ( L )( R )’, if any, is closedunder ⊔ , and forms, then, a(n additive) group of permutations. Also requiring that ∀ R, S (cid:2) ( L R ∧ L S ) ⇒ (cid:2) ð ( L ) (cid:0) R ⊔ S − (cid:1) ∧ ð ( L ) (cid:0) R − ⊔ S (cid:1)(cid:3)(cid:3) just amounts to requiring it. If L is a second-level (monadic) property such that L ( L ), thefirst-level binary relations satisfying the open formula ‘ ð ( L )( R )’, if any, form, then, a(n additive) One should better say ‘correspond to permutations’, since, strictly speaking, permutations are functions, notrelations. Let us adopt, however, a more straightforward, though abusive, language, for short. ⊔ , whose neutral element is theidentity permutation, and whose inverse function is R R − .This group is not necessarily Abelian, for ⊔ is not commutative on permutations. But itis endowed with a total and right-invariant order defined in terms of the composition operation.Since, if H and K are two permutations whatsoever that satisfy ‘ ð ( L ) ( R )’, requiring that H ⊔ K − have the property L is equivalent to requiring that K and H bear a right-invariant strict-orderrelation, let as say ❁ L , on these permutations . Hence, if this relation is conceived of as thesmaller-than relation (that is, ‘ L H ⊔ K − ’ or ‘ K ❁ L H ’ are read as ‘ K is smaller than H ’), thenwe can take the collection of the permutations that have L , if any, as the positive semi-group ofthe group of permutations formed by the permutations that satisfy ‘ ð ( L )( R )’ . Informally speaking, a nonempty positive class is a positival class whose domain is a totally-ordered, dense and Dedekind-complete group of permutations, which is, by consequence, also The proof is simple. As it has been required that ¬ L H ⊔ H − , we immediately have that ¬ H ❁ L H . As K ⊔ H − is the same permutation as (cid:0) H ⊔ K − (cid:1) − , we have that L H ⊔ K − ⇒ ¬ L K ⊔ H − , i.e. K ❁ L H ⇒ ¬ H ❁ L K .Again, if J is, also, a permutation that satisfies ‘ ð ( L ) ( R )’, then J ⊔ K − is the same permutation as (cid:0) J ⊔ H − (cid:1) ⊔ (cid:0) H ⊔ K − (cid:1) , and so we have that (cid:0) L H ⊔ K − ∧ L J ⊔ H − (cid:1) ⇒ L J ⊔ K − , i.e. ( K ❁ L H ∧ H ❁ L J ) ⇒ K ❁ L J .Finally, as ( H ⊔ J ) ⊔ ( K ⊔ J ) − is the same permutation as H ⊔ K − , we have that L H ⊔ K − ⇒ L ( H ⊔ J ) ⊔ ( K ⊔ J ) − ,i.e. K ❁ L H ⇒ K ⊔ J ❁ L H ⊔ J . In commenting his definition of positival classes, Frege (1893-1903, § II.175; Frege 2013, pp. 171 -72 ) claimsto have “tried [ bem¨uht ]” to include in it only “necessary [ nothwendigen ]” and “mutually independent [ einanderunabh¨angig ]” conditions, though taking as unprovable his having succeeded in this. In a note added at the end ofhis book ( ibidem vol. 2, p. 243, Frege 2013, p. 243 ), explicitly referred to this comment, he corrects himself byobserving that a proof could have been possible by means of counterexamples, though taking it to be “doubtful[ bezweifeln ]” that these counterexamples could be given in his formal setting. Dummett (1991, p. 288) suggests thathis doubt concerned the independence of the condition we expressed by ‘ ∀ R, S (cid:2) ( L R ∧ L S ) ⇒ ð ( L ) (cid:0) R − ⊔ S (cid:1)(cid:3) ’from the other ones characterizing a positival class, by observing that, in his developments concerning domainsof magnitudes, Frege appeals to this condition as late as possible (namely only in § II.218), after making explicit( § II.217) the “indispensability” of this condition for the purpose for which it is used, which, in our setting,corresponds to prove that if H and K belong to a positive class and H is smaller than K over the positivesemigroup involved in this class, then K − is smaller than H − over the corresponding group. Adeleke, Dummettand Neumann (1987, th. 2.1) have finally proved that this condition is actually independent of the others. Whentransposed in our setting, the proof goes along the following lines. Let L ⋆ be a property satisfying ‘ L ( X )’ exceptfor the condition at issue, G ⋆ be the structure formed by the permutations that satisfy ‘ ð ( L ⋆ ) ( R ), and H and K two binary first-level relations having L ⋆ . Insofar as (cid:0) K − ⊔ H (cid:1) − is the same permutation as H − ⊔ K , notensuring that K − ⊔ H satisfies ‘ ð ( L ⋆ ) ( R )’ is the same as not ensuring that the disjunction‘ L K − ⊔ H ∨ ∀ x, y (cid:2) x H − y ⇔ x K − y (cid:3) ∨ L H − ⊔ K ’ i.e. ‘ H − ❁ L K − ∨ H − = L K − ∨ K − ❁ L H − ’holds, namely that H − and K − are comparable according to the order over G ⋆ . It would follow that, besides ofnot being a group, this last structure is not endowed with a total order, but only with a partial one. It can beproved ( ibidem , Lemma 1.2) that this partial order is an “upper semilinear order”—that is, a strict partial order“such that the elements greater than any given one are comparable, and that, for any two incomparable elements,there is a third greater than both of them”, or, more simply, a strict partial order that “may branch downwards,but cannot branch upwards” (Dummett 1991, p. 288). But G ⋆ is a sub-structure of the group G formed by thepermutations that satisfy ‘ ð ( L ) ( R ), where L satisfies ‘ L ( X )’ as a whole. Hence, the condition at issue followsfrom the others if and only if G ⋆ can be extended in no group other than G . To prove the independence of thiscondition it is, then, enough to prove that there is a group including G ⋆ other than G . By Cayley’s theorem, anygroup is isomorphic to a group of permutations. It is, then, enough to prove that there is a partially ordered groupwhatsoever not isomorphic to G (that is, not totally ordered) that includes a sub-structure isomorphic to G ⋆ . Thisis just what Adeleke, Dummett and Neumann do. . By having a strict order available, the density condition can bestated easily. For stating the Dedekind-completeness one, further means are required.To this purpose, Frege defines the upper rims over a positival class. Let L be such that L ( L ),and A a sub-property of it. In our setting, an upper rim U of the collection of permutationsthat have A over the collection of those that have L is a relation having L , such that any otherrelation that has L and is smaller than U over the former collection has A . To define it, Fregebegins with a general definition, then applies it to positival classes. In the general case, both theinformal notion of an upper rim and the subsequent one of an upper limit become nonsensical.Formally speaking, this is immaterial, however, since the following definition of positive classesexcludes that the deviant cases obtain in the case of such a class .Once again, there are two ways to render the general definition: either as(3.3) ∀ X , Y ∀ R (cid:2) X ` R Y ⇔ ∀ S (cid:2)(cid:0) X S ∧ X R ⊔ S − (cid:1) ⇒ Y S (cid:3)(cid:3) , or as(3.3 ′ ) [( X ) ` ( Y )] ( R ) := ∀ S (cid:2)(cid:0) X S ∧ X ( R ⊔ S − (cid:1) ) ⇒ Y S (cid:3) . The upper limit of a sub-class of a positival class is the greatest of all the upper rims of theformer over the latter, if there is one. Anew, Frege’s definition can be rendered in two ways:either as(3.4) ∀ X , Y ∀ R (cid:2) X l R Y ⇔ (cid:8) L X ∧ X R ∧ X ` R Y ∧ ¬∃ S (cid:2) X S ∧ X S ⊔ R − ∧ X ` S Y (cid:3)(cid:9)(cid:3) , or as:(3.4 ′ ) [( X ) l ( Y )] ( R ) := ( L ( X ) ∧ X R ∧ [( X ) ` ( Y )] ( R ) ∧ ¬∃ S " X S ∧ X ( S ⊔ R − ) ∧ [( X ) ` ( Y )] ( S ) . For short, read both ‘ X ` R Y ’ and ‘[( X ) ` ( Y )] ( R )’ as ‘ R is an upper rim of Y over X ’, andboth ‘ X l R Y ’ and ‘[( X ) l ( Y )] ( R )’ as ‘ R is the upper limit of Y over X ’.Let P be a third-level monadic property of first-order binary relations. Informally speaking,the relations that have it form a positive class if they form a positival one, and are such that:for any relation R which has P , there is another relation S smaller than it over P (density);any proper subclass Y of P which has an upper rim over P also has an upper limit over P (Dedekind-completeness). These conditions can be rendered in two ways: either as(3.5) ∀ X P X ⇔ L X ∧∀ R [ X R ⇒ ∃ S [ X S ∧ X R ⊔ S − ]] ∧∀ Y "" ∃ R [ X ` R Y ∧ X R ] ∧∃ S [ X S ∧ ¬ Y S ] ⇒ ∃ T [ X l T Y ] , That a totally-ordered, dense and Dedekind-complete group (of permutations) is also Archimedean and Abelianis in fact proved by Frege himself. He proves that a Dedekind-complete positival class is Archimedean (Frege 1893-1903, th. 635, § II. 214), and that the domain of a positive class is Abelian ( ibidem , th. 689, § II. 244). This isthe last theorem he proves. Insofar as the proof of the former theorem does not appeal to the condition consideredin footnote (18) above, Adeleke, Dummett and Neumann (1987, p. 516) restate these theorems as follows: aDedekind-complete upper semilinear order is Archimedean—which, of course makes it also a Dedekind-completetotal order; if the order of a group is dense, Archimedean and total, then the group is Abelian. See footnote 21 below.
11r as:(3.5 ′ ) P ( X ) := L ( X ) ∧∀ R [ X R ⇒ ∃ S [ X S ∧ X R ⊔ S − ]] ∧∀ Y "" ∃ R ([( X ) ` ( Y )] ( R ) ∧ X R ) ∧∃ S ( X S ∧ ¬ Y S ) ⇒ ∃ T [[( X ) l ( Y )] ( T )] . For short, read both ‘ P X ’ and ‘ P ( X )’ as ‘ X is a positive class’ or, more precisely, ‘thefirst-level binary relations having X form a positive class’.While (3.3), (3.4) and (3.5) are explicit definitions, and have to be licensed by some form offourth-order comprehension, (3.3 ′ ), (3.4 ′ ) and (3.5 ′ ) are abbreviation stipulations, and require noform of comprehension stronger than (PCA ) .From the previous remarks, it should be clear that if the second-level monadic property P is such that P ( P ), then the permutations that respectively satisfy ‘ ð P R ’ or ‘ ð ( P ) ( R )’, if any,form a totally-ordered, dense and Dedekind-complete group of permutations. This is just what a Comparing (3.5) and (3.5 ′ ), on the one side, with (3.3-3.4) and (3.3 ′ -3.4 ′ ), on the other, allows us to see whythe deviant cases pertaining to the definition of an upper rim and an upper limit become immaterial by passingto the definition of a positive class. For short and simplicity, we only consider (3.3 ′ -3.5 ′ ). The right-hand sideof (3.3 ′ ) fails, as such, in rendering the informal notion of an upper rim since it does not express the two crucialconditions that Y be a sub-class of X , and that X be a positival class and R belong to it. This makes ‘ X R ⊔ S − ’do not render the condition that S be smaller than R over X . Hence, according to (3.3 ′ ), it could happen that[( L ) ` ( H )] ( R ) even if H is not a sub-class of L or R it is not such that any relation smaller than it over L has H . It follows that the conjunction L ( L ) ∧ L R ∧ [( L ) ` ( H )] ( R )renders the informal condition that R be an upper rim of H over L except for the requirement that H be asub-class of L . What are the consequences of missing this requirement? To see it, let us write the implication‘ ∀ S (cid:2)(cid:0) X S ∧ X R ⊔ S − (cid:1) ⇒ Y S (cid:3) ’ as ‘ ¬∃ S (cid:2) X S ∧ X R ⊔ S − ∧ ¬ Y S (cid:3) ’ . It this clear that this formula can be satisfied by L (as value of X ) and H (as value of Y ) even if H is not asub-class of L . For instance, this is just what happens, whatever first-level binary relation R might be, if L is asub-class of H . Hence, missing the mentioned requirement results in admitting that, for any R , if L is a sub-classof H , then [( L ) ` ( H )] ( R ). But suppose that the first-level binary relations that have L form a positival classand that R be one of them, that is, that L ( L ) ∧ L R . If R is not the smallest relation that has L , there iscertainly another relation S such that L S ∧ L R ⊔ S − . Hence, for it to hold that ¬∃ S (cid:2) L S ∧ L R ⊔ S − ∧ ¬ H S (cid:3) and [( L ) ` ( H )] ( R ) , it is necessary that any such S have H . But if this is so, then L and H are not disjoint. This having beenestablished, rewrite the right-hand side of (3.4 ′ ) in agreement with (3.3 ′ ), i.e. as follows L ( X ) ∧ X R ∧ ¬∃ S (cid:2) X S ∧ X R ⊔ S − ∧ ¬ Y S (cid:3) ∧ ¬∃ T (cid:2) X T ∧ X T ⊔ R − ∧ ¬∃ W (cid:0) X W ∧ X T ⊔ W − ∧ ¬ Y W (cid:1)(cid:3) . For this conjunction to hold, it has to exist a first-order binary relation that has X but not Y . The case where L is a sub-class of H is then expunged from those in which it can happen that [( L ) l ( H )] ( R ) for some R . Insofaras (3.4 ′ ) implies that [( L ) l ( H )] ( R ) only if L is positival, R has it, and [( L ) ` ( H )] ( R ), it follows that, providedthat R be not the smallest relation having L , it can happen that [( L ) l ( H )] ( R ) only if H is a sub-class of L or, at least, L and H are not disjoint, but L is not a sub-class of H . Let now P be a property of first-levelbinary relations satisfying the right-hand side of (3.5 ′ ), and, then, such that P ( P ). The sub-group formed by therelations having it is, then, dense. If R has P , it cannot happen that it be the smallest relation having it. Hence,it can happen that [( P ) ` ( H )] ( R ), only if H is either a subclass of P , or P and H are not disjoint, but P isnot a subclass of H . Hence P and H are not disjoint, that is, some relation having H has also P . Thus, evenif H is not a sub-class of P , it can happen that it has both some upper rims and an upper limit over P , whichis just what is relevant for both (3.4 ′ ) and (3.5 ′ ) to comply with the informal explanations given above. ∀ X [ M X ⇔ ∃ Y [ P Y ∧ ∀ R [ X R ⇔ ð Y R ]]] , or the following abbreviation stipulation requiring no comprehension stronger than (PCA ),(3.6 ′ ) M ( X ) := ∃ Y [ P ( Y ) ∧ ∀ R [ X R ⇔ ð ( Y ) ( R )]] . where both ‘ M X ’ and ‘ M ( X )’ are to be read as ‘ X is a domain of magnitudes’ or, moreprecisely, ‘the first-level binary relations having X form a domain of magnitudes’. As a matter of fact, (3.6) and (3.6 ′ ) provide two different definitions of domains of magnitudes.The former results from the explicit definition of the third-order predicate constant ‘ M ’. Thelatter merely exhibits the third-order open formula briefly designated by ‘ M ( X )’, and involvesno explicit definition other than (Inversion) and (Composition). Both render Frege’s definition,but require different logical resources, and play distinct roles in our setting.Let us begin with the logical resources they require. A first difference is manifest: while(3.6) requires a fourth-order system, a third-order system is enough for (3.6 ′ ). Though bothsystems encompass no proper axioms, the former is, by far, more entangled than the latter. Thisis not only because of its higher order, but also because of the forms of comprehension it hasto incorporate, in order to license (3.6). Besides (PCA )—or its instances (Composition-CA)and (Inversion-CA)—, required to license (Composition) and (Inversion), it also calls for othercomprehension axioms, respectively required to license (Functionality) and (3.1-3.6) . The lattersystem only needs to involve (PCA ), or merely (Composition-CA) and (Inversion-CA), instead.First-order variables (and the usual logical constants) apart, the language of this latter systemhas only to include dyadic second-order and monadic third-order variables, together with the Namely:(CA ) ∃ X ∀ R h X R ⇔ φ ∆ i [where ‘ φ ∆ ’ stands for a second-order predicative formula],required to license (Functionality);(PCA ) ∀ X ∃ Y ∀ R (cid:20) Y R ⇔ φ Σ ( X ) (cid:21) where ‘ φ Σ ( X )’ stands for a third-order formulainvolving a second-order existential quantifierand the parameter ‘ X ’ , required to license (3.1);(CA ) ∃X ∀ X (cid:20) X X ⇔ φ Π (cid:21) " where ‘ φ Π ’ stands for a third-order formulainvolving a second-order universal quantifier , required to license (3.2);(CA ) ∃V∀ X , Y ∀ R (cid:20) X V R Y ⇔ φ Π (cid:21) [where ‘ φ Π ’ is as in (CA )],required to license (3.3); , and the functional constants ‘ − ’ and ‘ ⊔ ’. Though third-order, thissystem is, then, quite weak. As a matter of fact, we have nevertheless shown that Frege’s definitionof domains of magnitudes can be consistently rephrased in such a weak system, and is, then, sorephrased, equiconsistent with it . For future reference, call this system ‘L PCA ’.It remains, however, that (3.6 ′ ), and, then, this very system, are suitable for our presentpurpose only if we are content with admitting that a second-level property M is a domain ofmagnitudes (or that the first-level binary relations that have it form such a domain) if and onlyif it satisfies the right-hand side of (3.6 ′ ). Were we, instead, in need of defining a (third-level)property that M has if and only if it is so, (3.6 ′ ) would no more be suitable, and we would have torecur to (3.6) and the corresponding fourth-order and much stronger system. Provided that thedefinition of positival and positive classes is, in the present setting, merely instrumental to thatof domains of magnitudes, the former attitude might be easily admitted for the correspondingproperties. But one might consider that the same attitude is not admissible in the case of thesevery domains, whose definition is the final outcome of Frege’s work, on pain of missing a genuineentity counting as such a domain, and, then, the definition itself.Still, even if the definition of domains of magnitudes is the last step Frege reached in hisformalization of real analysis , it is in no way the final step such a formalization should havereached: this should have rather been an explicit definition of real numbers, and, possibly, of theoperations and relations making them form a totally ordered and Dedekind-complete field. Hence,if Frege’s informal indications for reaching this final aim may be rendered in our setting withoutdefining the predicate constant ‘ M ’, there is no stringent reason for accepting the foregoingargument, so that (3.6 ′ ) and L PCA may be considered enough for the purpose of renderingthe result he was envisaging. In §
5, we will show that this can be actually done. We can, then,conclude that, whereas (3.6), and the fourth-order system it requires, are suitable for the purposeof defining domains of magnitude, as such, (3.6 ′ ) and L PCA are so for the purpose of definingreal numbers as ratios on such domains. As this is our goal, we’ll go, then, for the latter option.This is all the more justified because no form of comprehension can guarantee the existence (CA ) ∃V∀ X , Y ∀ R (cid:20) X V R Y ⇔ φ Σ (cid:21) " where ‘ φ Σ ’ stands for a fourth-order formulainvolving a second-order existential quantifier , required to license (3.4);(CA ) ∃X ∀ X (cid:20) X X ⇔ φ Π (cid:21) " where ‘ φ Π ’ stands for a fourth-order formulainvolving a third-order universal quantifier , required to license (3.5);(CA ) ∃X ∀ X (cid:20) X X ⇔ φ Σ (cid:21) " where ‘ φ Π ’ stands for a fourth-order formulainvolving a third-order existential quantifier , required to license (3.6). Notice that a third-order quantifier only occurs once: in the right-hand side of (3.5 ′ ). This single occurrenceis however essential for the definition of domains of magnitude to succeed. A note of caution. The fact that, when rephrased as suggested, Frege’s definition is equiconsistent with thissystem does not entail at all that the original definition requires no impredicative comprehension, and is, then,consistent in itself. It crucially involves the function ξ ⌢ ζ , which cannot be defined without impredicative(second-order) comprehension. To be more precise, Frege offers no explicit formal definition of domains of magnitudes, and rather is contentwith informally claiming that a domain of magnitudes is the domain of a positive class.
14f positival and positive classes and domains thereof. Surely, by the standard interpretation ofhigher-order logic, the stipulations (3.2 ′ ) and (3.5 ′ -3.6 ′ ) being given, the following instances ofthird-order impredicative comprehension ∃ X ∀ R [ X R ⇔ ∃ Y ( L ( Y ) ∧ Y R )] , ∃ X ∀ R [ X R ⇔ ∃ Y ( P ( Y ) ∧ Y R )] , ∃ X ∀ R [ X R ⇔ ∃ Y ( M ( Y ) ∧ Y R )]are enough to ensure the existence of the second-level properties that a first-level binary relationhas to have for being respectively included in a positival and a positive class and in a domain ofmagnitudes. Again, the following instances of fourth-order predicative comprehension ∃X ∀ X [ X X ⇔ L ( X )] , ∃X ∀ X [ X X ⇔ P ( X )] , ∃X ∀ X [ X X ⇔ M ( X )]are enough to ensure the existence of the third-level properties of being a positival and a positiveclass and a domain of magnitudes. Still, securing this existence is no substantial achievement,since these properties would exist even if they were empty, or there were not enough relationssatisfying them, to make them play the required role in a definition of the reals.Even more so, if we grant the extensional conception of properties and relations, we also haveto grant that, no matter how the first-order domain might be, both the empty first-level binaryrelation and the empty second-level property exist , which alone is enough to ensure that allsix foregoing second- and third-level properties exist, in turn, and are nonempty, since, accordingto (3.2 ′ ) and (3.5 ′ -3.6 ′ ), the empty second-level property is at the same time a positival class, apositive one, and a domain of magnitudes . Frege is not clear about what he takes to make a property or a relation exist, if at all. It is even plausible toascribe him an intensional conception, which makes the talk of existence of properties and relations inappropriate(Panza 2015). What is unquestionable is that he does not explicitly endorse any sort of comprehension axiom, byrather pervasively admitting of a substitution policy which we could consider as equivalent to full second-ordercomprehension. In the light of BLV, our replacing value-ranges of binary first-level relations with these veryrelations seems, however, in line with granting the extensional conception of properties and relations while doingsemantic considerations about L PCA . This means informally taking a n + 1-order m -adic predicate variableto range on all the subsets of ordered m -tuples of the elements of the n -order domain ( n, m = 1 , , . . . ), and apredicate constant to designate one of these subsets. Let E be the second-level empty property. From (3.2 ′ ) it follows that L ( E ), since ∀ R ¬ [ E R ]. Let V bethe empty first-level binary relation, and E the second-level property of being this property. If R is a first-level binary relation (extensionally) distinct from V , then, we have that both E V and ¬ E R . Moreover, from(Funcionality), (Inversion ′ ) and (Composition ′ ), it follows that I ( V ), and that V extensionally coincides bothwith V − and with V ⊔ V − , and, then, also with V − ⊔ V (see Frege, 1893-1903, § II.164). This is enough toshow that V does belong to no positival (and, then, positive) class, and that both ¬L ( E ) and L ∗ ( E ), where‘ L ∗ ( X )’ is the abbreviation of the formula resulting from the right-hand side of (3.2 ′ ), by skipping the conjunct‘ ¬ X R ⊔ R − ’. Then, E is positival, except for admitting the possibility that the identity relation be includedin it. Thus, both the second-level properties [ R : ∃ Y ( L ( Y ) ∧ Y R )] and [ R : ∃ Y ( L ∗ ( Y ) ∧ Y R )] and the thirdlevel ones [ X : L ( X )] and [ X : L ∗ ( X )] not only exist by appropriate forms of comprehension, but are alsononempty. Consider now (3.5 ′ ). It is enough to observe that L ( E ) and ¬∃ R [ E R ] to conclude that P ( E ).Look, then, at E . Insofar as R ⊔ V − coincides with V , whatever first-level binary relation R might be, from(3.3 ′ ) it follows that [( E ) ` ( Y )] ( R ) if and only if Y is E itself. Insofar as ∀ S ¬ [ E S ∧ ¬ E S ], from (3.5 ′ ) it also . And, if this holds for a positive class, it also holds for its domain. Itfollows that a positive class and a domain of magnitudes exist, which include at least a first-levelbinary relation, if and only if there exists one including continuously many of them. No semanticconsideration is, however, apt to prove the existence of nonempty positive classes and domainsthereof. To prove it, it is necessary to prove that there are enough objects, for defining on themcontinuously many appropriate relations.This was the crucial challenge for Frege’s definition of the real numbers, and it is also ours.Before tackling it, it is, however, in order to see what makes this proof indispensable both forFrege’s and for our purposes. This requires looking at how Frege’s original theory of the reals asobjects can be revived in our setting, on the basis of (3.5 ′ ). Apart from some generalities, most of which we already discussed above, and the sketchy outlineof a possible existence proof for nonempty domains of magnitudes, which we will consider in § pars construens of part III of Grundgesetze only contains the definition of these domains. Noprecise indication of how to define the real numbers is available. The only thing that is clear isthat these numbers should be defined as ratios of magnitudes, and that these ratios have to be follows that P ∗ ( E ), where ‘ P ∗ ( X )’ is the abbreviation of the formula resulting from the right-hand member of(3.5 ′ ) by replacing ‘ L ( X )’ with ‘ L ∗ ( X )’. So, the properties [ R : ∃ Y ( P ( Y ) ∧ Y R )], [ R : ∃ Y ( P ∗ ( Y ) ∧ Y R )],[ X : P ( X )] and [ X : P ∗ ( X )] not only exist by appropriate forms of comprehension, but they are also nonempty.Finally, consider (3.6 ′ ). Provided that ‘ M ∗ ( X )’ be the abbreviation of the formula resulting from the right-handside of (3.6 ′ ) by replacing ‘ P ( X )’ with ‘ P ∗ ( X )’, it is immediate that both M ( E ) and M ∗ ( E ), just because P ( E ) and P ∗ ( E ) and the domains of E and E respectively coincide with E and E themselves. Hence, theproperties [ R : ∃ Y ( P ( Y ) ∧ Y R )], [ R : ∃ Y ( P ∗ ( Y ) ∧ Y R )], [ X : M ( X )] and [ X : M ∗ ( X )], too, not only existby appropriate forms of comprehension, but are also nonempty. Notice that M ∗ does not (extensionally) coincidewith M , since M ∗ ( E ), but not M ( E ). More precisely, this is with respect to the full model of L PCA , which we also take as its intended one,where the first-order variables, x , y , z , . . . vary over a large enough domain A of objects, the second-order dyadicvariables R , S , T , . . . vary over the full power set of A × A , and the third-order monadic variables X , Y , . . . varyover the full power set of the power set of A . It is, indeed, easy to see that L PCA is far from categorical. Asimple way to see it (thanks to Andrew Moshier for suggesting it to us) is to observe that the lack of third-ordercomprehension makes this system have a model where the third-order variables vary over the empty set. In thismodel, all closed formulas beginning by a third-order universal quantifier, as the third conjunct in the right-handside of (3.5 ′ ), are vacuously true. This makes ‘ P ( X )’ trivially satisfied by countably many (appropriate) binaryrelations (whereas PCA makes any model of L PCA include countably many first-level binary relations).One might be surprised we take the full model to be the intended model, rather than an appropriate Henkinone. Still, we think a restriction on comprehension, or even the lack of it, is no reason for imposing a restrictedsemantics: one thing is the logical resources, in particular the instances of comprehension, required for a definition;another the selection of the intended model. The former are deductive, syntactical tools required to formulatedefinitions; the latter depends on semantic considerations relative to the informal piece of knowledge that therelevant system is expected to render—which, in this case, involves the idea of a positive class as a complete semi-group of permutations. We are not going to take a stand on this matter, here, but costs and benefits are worthmentioning. Should our definition be required to recover positive classes including exactly 2 ℵ permutations, thenthe intended semantic is to be clearly full. If impredicativity were deemed a price worth paying for this purpose,one should also be aware of the heavy-duty logical resources called for it. . A simple way to accomplish this plan is by an abstraction principle governing an operator takingpairs of relations (i.e. magnitudes) from a domain of magnitudes as arguments and having objects(i.e. ratios) as values. As suggested by Simons (1987, pp. 39–40) and Dummett (1991, pp. 290–91), this can be done by rephrasing definition V.5 of Euclid’s
Elements , and defining the relationof proportionality between four magnitudes taken two by two This raises a technical difficulty, however. Whereas this definition applies only if the magni-tudes composing each pair are such that it cannot be the case that any multiple of one of thembe smaller, equal, or greater than any multiple of the other, this condition is not met by pairs ofmagnitudes of the same domain, since, differently from what happened for Euclid’s ones, theseare intended to be either positive, negative, or null. A way to solve the difficulty is by appropri-ately modifying the very structure of Euclid’s definition, in order to get the following abstractionprinciple, which deserves, nevertheless, the name of ‘Euclid’s Principle’ (or ‘EP’, from now on),(EP) ∀ ( P ) X , X ′ ∀ ( ð ( X )) R ∀ X S ∀ ( ð ( X ′ )) R ′ ∀ X ′ S ′ R [ R, S ] = R [ R ′ , S ′ ] ⇔ " X R ∧ X ′ R ′ ∧∀ N x, y ( xR ❁ X yS ⇔ xR ′ ❁ X ′ yS ′ ) ∨ " X R − ∧ X ′ R ′− ∧∀ N x, y ( xR − ❁ X yS ⇔ xR ′− ❁ X ′ yS ′ ) ∨ " ∀ z, w " ( zRw ⇔ z [ S ⊔ S − ] w ) ∧ ( zR ′ w ⇔ z [ S ′ ⊔ S ′− ] w ) , where:– ‘ R ’ is the relevant abstraction operator;– ‘ ∀ ( P ) X [ ϕ ]’ abbreviates ‘ ∀ X [ P ( X ) ⇒ ϕ ]’, and the same for ‘ X ′ ’; From what he writes in the very § of his treatise, it seems that Frege was also requiring that real numbersform themselves a domain of magnitudes (Frege 1893-1903, § II.245; Frege 2013, p. 243 ):The commutative law for the domain of a positive class is thus proven. The next task is now toshow that there is a positive class, as indicated in §
164 [see p. 2, above]. This opens the possibilityof defining a real number as a ratio of magnitudes of a domain that belongs to a positive class.Moreover, we will then be able to prove that the real numbers themselves belong as magnitudes tothe domain of a positive class.This further requirement would have not only uselessly entangled Frege’s own first-order definition, if he completedit (Dummett 1991, pp. 190-91), but it is also logically incompatible, in our predicate setting, with the requirementthat real numbers be objects. This is why we will set it aside in our reconstruction. As observed by Dummett (1991, pp. 282–83), Euclid’s definition, probably tracing back to Eudoxus, in fact,had been explicitly appealed to by H¨older in his paper on the “axioms of quantity” (H¨older 1901), which appearedtwo years before the second volume of the
Grundgesetze . But, apparently, Frege’s was not aware of this. On H¨older(1901), cf. B laszczyk (2013), which also sums up how the notion of magnitude was investigated around the end of19th century by several mathematicians, including Du Bois-Reymond, Stolz, and Weber, by explicitly referring toEuclid’s theory, and achieving results mathematically equivalent to Frege’s.
17 ‘ ∀ ( ð ( X )) R [ ϕ ]’ abbreviates ‘ ∀ R [ ð ( X ) ( R ) ⇒ ϕ ]’, and the same for ‘ X ′ ’ and ‘ R ′ ’;– ‘ ∀ X S [ ϕ ]’ abbreviates ‘ ∀ S [ X R ⇒ ϕ ]’, and the same for ‘ X ′ ’ and ‘ S ′ ’;– ‘ ∀ N x [ ϕ ]’ abbreviates ‘ ∀ x [ N x ⇒ ϕ ]’, and the same for ‘ y ’;– ‘ N ’ denotes the property of being a natural number;– ‘ xR ’ abbreviates ‘ R ⊔ R ⊔ . . . ⊔ R | {z } x times , and the same for ‘ y ’, ‘ S ’, ‘ S ′ ’, R ′ ’, ‘ R − ’, ‘ R ′− ’;– ‘ xR ❁ X yS ’ abbreviates ‘ X yS ⊔ ( xR ) − ’, and the same for ‘ X ′ ’, ‘ R ′ ’, ‘ S ′ ’, ‘ R − ’, ‘ R ′− ’.Informally speaking, EP states that for whatever pairs of domains of magnitudes, issued bytwo positive classes P and P ′ , and whatever two ordered pairs of permutations R , S and R ′ , S ′ , such that R , and R ′ respectively belong to the domains of these classes, while S and S ′ belongto the classes themselves (and are, then, intended as positive), the ratio R [ R , S ] of the elementsof the first pair is the same as the ratio R [ R ′ , S ′ ] of the elements of the second pair if and only if:– either both the first elements R and R ′ of these pairs belong to the respective positiveclasses P and P ′ (and are, then, intended as positive), and their equimultiples are alwayssmaller than the equimultiples of the second elements ;– or both the inverses R − and R ′− of the first elements R and R ′ of these pairs belong tothe respective positive classes P and P ′ (so that R and R ′ are intended as negative), andtheir equimultiples are always smaller than the equimultiples of the second elements;– or both the first elements R and R ′ of these pairs are the identity relation (and are, then,intended as null).So rephrasing Euclid’s definition surely solves the technical difficulty, but it does not solveall problems: though EP involves neither a predicate constant ‘ P ’ for the third-level property ofbeing a positive class, nor a functional constant ‘ ð ’ for the domain operator, but merely dependson the stipulations (3.5 ′ ) and (3.1 ′ ), it cannot, as such, be added to L PCA as a new axiom,so as to get an extended system in which real numbers are to be defined. There are two reasonsfor that. First of all, EP involves the predicate constant ‘ N ’ for the first-level property of beinga natural number, which is not and cannot be defined within L PCA . Secondly, it involvesthe symbol ‘ xR ’ (or ‘ R ⊔ R ⊔ . . . ⊔ R | {z } x times ’) (where ‘ x ’ is a variable ranging over the natural numbers)whose use in a formal system is licensed only if this latter contains a device to count the iteratedapplications of the functional constant ‘ ⊔ ’, which is not and cannot be provided within L PCA .A way to overcome the first issue is to extend L PCA to a stronger system, in which theproperty of being a natural number can somehow be defined, e.g. by adding HP as a new axiom Notice that EP does not involve domains of magnitudes as such, but rather positive classes and their domains.This is perfectly in line with Frege’s missing a formal definition of these domains: see footnote (25), above. Notice also that, whereas Euclid’s definition requires that the equimultiples of the first and the third, amongthe four relevant magnitudes, “alike exceed, are alike equal to, or alike fall short of [ ἅμα ὑπερέχῃ ἢ ἅμα ἴσα ᾖ ἢ ἅμαἐλλείπῃ ]” (Euclid EH, vol. II, p. 114) the equimultiples of the second and the fourth, we can just require that theequimultiples of the first and the third magnitudes all be smaller than those of the second and the fourth, since,as noticed by Scott (1958-59), in the case of Archimedean magnitudes, the latter condition entails the former. §§ II.199-214). It consists in amending EP with the help ofsome new abbreviation stipulations, which merely require adding new third-order binary variables.
Let us begin by adopting the following new abbreviation stipulation: D ( T ) ( R, S ) := ∀ x, y ( xSy ⇔ x [ T ⊔ R ] y ) . Let R , S , and T , be whatever first-level binary relations. According to this stipulation, theformula ‘ D ( T ) ( R , S )’ asserts that S results from, or extensionally coincide with, the compositionof T and R . Hence, ‘ D ( R ) ( R , S )’ asserts that S results from the composition of R with itself. Inthe notation employed in stating EP, this means that S coincides with 2 R .This allows to simulate the usual definition of the weak ancestral of a binary relation: D ⊔⊔ ( R ) ( S ) := ∀ X X R ∧∀ R ′ , S ′ "" X R ′ ∧ D ( R ′ ) ( R ′ , S ′ ) ⇒ X S ′ ⇒ X S . This makes the formula ‘ D ⊔⊔ ( R ) ( S )’ assert that S extensionally coincides with R or with the relationresulting from a iterated composition of R with itself, and is, then, a multiple of R itself. In thenotation employed in stating EP, this means that S coincides with n R , for some natural number n . Let, now, P be a positive class, and R a relation in it. It is clear that if D ⊔⊔ ( R ) ( T ) and D ⊔⊔ ( R ) ( S ),then both T and S belong to P . Suppose it is so, and that P T ⊔ S − . We can, then, take S to be smaller than T over P . Hence, if also P ′ is a positive class (either distinct from P ornot), R ′ is a first-level binary relation that belongs to it, and it is also the case that D ⊔⊔ ( R ′ ) ( T ′ )for some first-level binary relation T ′ , then T is the same multiple of R over P as T ′ of R ′ over P ′ if and only if there are as many first-level binary relations that satisfy the open formula‘ D ⊔⊔ ( R ) ( S ) ∧ P T ⊔ S − ’ as those that satisfy the other open formula ‘ D ⊔⊔ ( R ′ ) ( S ′ ) ∧ P ′ T ′ ⊔ S ′− ’.This suggests enriching the language of L PCA by introducing third-order binary variables,ranging over second-level binary relations between first-level such relations, and adopting the19ollowing further abbreviation stipulation(5.1) ( X , X ′ ) E ( R, T, R ′ , T ′ ) := D ⊔⊔ ( R ) ( T ) ∧ D ⊔⊔ ( R ′ ) ( T ′ ) ∧∃ R ∀ S (cid:16) D ⊔⊔ ( R ) ( S ) ∧ S ⊑ X T (cid:17) ⇒∃ ! S ′ h S R S ′ ∧ D ⊔⊔ ( R ) ( S ′ ) ∧ S ′ ⊑ X ′ T ′ i ∧∀ S ′ (cid:16) D ⊔⊔ ( R ′ ) ( S ′ ) ∧ S ′ ⊑ X ′ T ′ (cid:17) ⇒∃ ! S h S R S ′ ∧ D ⊔⊔ ( R ) ( S ) ∧ S ⊑ X T i , where ‘ R ’ is such a variable, and ‘ S ⊑ X T ’ abbreviates ‘ X T ⊔ S − ∨ ∀ x, y [ xSy ⇔ xT y ]’, and thesame as for ‘ X ′ ’, ‘ T ′ ’ and ‘ S ′ ’. Thus, if P , P ′ , R , R ′ , T , and T ′ are as above, then the formula‘ ( P , P ′ ) E ( R , T , R ′ , T ′ )’ asserts that T is the same multiple of R over P as T ′ of R ′ over P ′ .For short, let us, now, adopt this other abbreviation stipulation: ( X , X ′ ) E ( R,T,R ′ ,T ′ )( S,U,S ′ ,U ′ ) := ( X , X ′ ) E ( R, T, R ′ , T ′ ) ∧ ( X , X ′ ) E ( S, U, S ′ , U ′ ) . EP can, then, be restated as follows:(EP ∗ ) ∀ ( P ) X , X ′ ∀ ( ð ( X )) R ∀ X S ∀ ( ð ( X ′ )) R ′ ∀ X ′ S ′ R [ R, S ] = R [ R ′ , S ′ ] ⇔ X R ∧ X ′ R ′ ∧∀ T, U, T ′ , U ′ ( X , X ′ ) E ( R,T,R ′ ,T ′ )( S,U,S ′ ,U ′ ) ⇒ ( T ❁ X U ⇔ T ′ ❁ X ′ U ′ ) ∨ X R − ∧ X ′ R ′− ∧∀ T, U, T ′ , U ′ ( X , X ′ ) E ( R − ,T,R ′− ,T ′ )( S,U,S ′ ,U ′ ) ⇒ ( T ❁ X U ⇔ T ′ ❁ X ′ U ′ ) ∨ " ∀ zw [ zRw ⇔ z [ S ⊔ S − ] w ] ∧∀ zw [ zR ′ w ⇔ z [ S ′ ⊔ S ′− ] w ] . It should be easy to verify that, informally speaking, EP ∗ has the same content as EP. Butit expresses this content in the language of L PCA , merely enriched by the addition of third-order binary variables as ‘ R ’. This addition being admitted, EP ∗ can, then, be added to thissystem as a supplementary axiom. Since EP ∗ is an abstraction principle, its left-hand side is afirst-order identity (i.e. ‘ R [ R, S ]’ and ‘ R [ R ′ , S ′ ]’ are singular terms). Moreover, its right-handside involves no constant other than ‘ − ’ and ‘ ⊔ ’. Hence, adding it to L PCA requires nofurther comprehension axiom. The theory obtained is, then, a third-order one, including first-order, second-order binary, and third-order monadic and binary variables, but only admittingpredicative second-order comprehension. 20 .3 Real Numbers Though EP ∗ supplies the required grounds for defining real numbers as objects, this theory fallsshort of achieving it. All that one can do, in the light of it, is informally (and meta-theoretically)identify these numbers with ratios like R [ R , S ]. If a predicate constant designating the first-level property of being a real number is to be available, one also has to admit a new form ofcomprehension, for licensing the following explicit definition:(5.2) ∀ x (cid:2) R x ⇔ ∃ ( P ) X ∃ ( ð ( X )) R ∃ X S [ x = R [ R, S ]] (cid:3) . What we need, then, is the following second-order third-orderly impredicative axiom-scheme:(CA ) ∃ X ∀ x h Xx ⇔ φ Σ i , where ‘ φ Σ ’ stands for a third-order formula involving a third-order existential quantifier—together with a second-order one.It is then only in such a (highly) impredicative third-order theory obtained from L PCA byadding to it both the proper axiom EP ∗ and the comprehension axiom-schema (CA ), that theproperty of being a real number can be properly defined in our predicate setting. For short, letus call this theory ‘FMR’ (for ‘Frege’s (theory of) magnitudes (and) real (numbers)’). If such animpredicative theory were to be avoided, definition (5.2) should be omitted. At most, one couldrecur to a new abbreviation stipulation as(5.2 ′ ) R ( x ) := ∃ ( P ) X ∃ ( ð ( X )) R ∃ X S [ x = R [ R, S ]] , by then admitting that a real number is an object that satisfies the open formula ‘ R ( x )’. Call‘FMR ′ ’ the theory got from L PCA by merely adding EP ∗ , and replacing (5.2) with (5.2 ′ ). Thesame argument used above to prefer (3.6 ′ ) over (3.6) does not apply here, since the definition ofreal numbers is the final purpose to be reached to revive Frege’s program. Hence, if one considersthat this aim is reached only if a property, counting as the property of being a real number, isdirectly expressed as such, in the relevant formal setting, on pain of missing the definition itself,there is no other option than going for FMR.According both to (5.2) and (5.2 ′ ), a real number is a ratio over some domain of magnitudes.This might appear odd at first glance, since, given different such domains, this might seem toentail that different sorts of real numbers arise, according to the domain of magnitudes they aredefined on. However, from EP ∗ , it easily follows that, if there are several domains of magnitudes,for any ratio (or R - abstractum ) on one of them, there is just another ratio (or R - abstractum ) oneach other of them that is the same object as the former—i.e. that the ratio of two first-levelbinary relations having a certain property M such that M ( M ) is the same object as the ratioof two first-level binary relations having another property M ′ such that M ( M ′ ).Hence, once real numbers are defined, either in FMR or in FMR ′ , as ratios of magnitudes,one can define the usual properties, relations and functions on them, making the development ofreal analysis possible, within these systems—or some appropriate extensions of them, if needed.We stop here, however, and rather tackle some meta-theoretical issues concerning these systems,and the corresponding definitions. 21 Existence Proofs
It is easy to see that EP ∗ implicitly defines continuously many objects to be identified, eitherthrough (5.2) or through (5.2 ′ ), with the real numbers, only in the presence of nonempty positiveclasses. If there were no first-level binary relations R such that ∃ X [ P ( X ) ∧ X R ], its second-order universal quantifier would range on an empty domain, and this would render the right-handside of EP ∗ nonsensical, as well as, then, both (5.2) and (5.2 ′ ) Still, a nonempty positive classexists if and only if this is so for a nonempty domain of magnitudes. Hence, an existence proofof such a domain (or of a positive class) is an indispensable supplement to our definition of realnumbers: it is required to make it sensible.Of course, no form of comprehension might be appealed to in order to deliver this proof,since no comprehension axiom can secure the existence of an R such that ∃ X [ M ( X ) ∧ X R ].Moreover, it would not be enough to observe that the empty first-level binary relation exists nomatter what the first-order domain looks like, since, as observed in footnote (27), this relationneither forms nor belongs to a positival (and, then, positive) class. What is to be proved, then,is that there are enough appropriate (or appropriately related) objects for defining on themcontinuously many (extensionally) distinct first-level binary relations forming a nonempty domainof magnitudes .This cannot be accomplished by a proof following a similar pattern as the one that allowsneologicists to prove the existence of natural numbers within the very theory in which they definethem, namely FA. This proof goes as follows:– The concept [ x : x = x ] exists by predicative comprehension;– Then, HP allows to define 0 as x : x = x ];– By logic, [ x : x = x ] ≈ [ x : x = x ];– Hence, by HP, 0 = 0, from which it follows that 0 exists ;– Since HP allows to define the successor relation on the whole first-order domain, naturalnumbers can be defined as the objects that bear the weak ancestral of this relation to 0; This is just what Frege seems to signal at the beginning of § II.164, in the passage we have quickly referred toin footnote (27) above (Frege 1893-1903, § II.164; Frege 2013, pp. 160 -61 ; notice that the English term ‘Relation’with capital ‘R’ is used here to translate the German term ‘Relation’, which Frege uses, as opposed to ‘Beziehung’,translated instead as ‘relation’, to name value-ranges of relations):We can now approach the question raised earlier ( § q is theempty Relation, then [. . . ][ q − ] is the same; likewise [. . . ][ q ⊔ q − ]. Also the composition of Relationson our domain of magnitudes cannot result in the empty Relation; but that would happen if therewere no object to which some object stood in the first Relation and which also stood to some objectin the second Relation. We thus require a class of objects that stand to each other in the Relationsof our domain of magnitudes, and, in particular, this class has to contain infinitely many objects. Notice that, since HP licenses the formation of the term ‘ x : x = x ]’, the identity ‘0 = 0’ might be derived,in classical logic, as an immediate consequence of the theorem ‘ ∀ x [ x = x ]’. Still, if such a proof of the existence of0 were admitted, HP would inevitably be endowed with an existential import that would be incompatible with itsalleged analyticity. This is one of the reasons why it is often advanced that the subjacent logic to FA should befree: the matter has been firstly tackled in Shapiro & Weir (2000), §§ IV-V; but see also Payne (2013).
22 Proving—from HP plus (impredicative) comprehension—that any such object has a (unique)successor is, then, enough to prove, by countable induction, that all the natural numbersexist.This pattern only allows to prove that there are objects falling under a first-level concept,given both a way to identify these objects collectively, as values of a particular function such as —the main task of the proof would just be to prove that, which iscertainly not something that might be done by following this pattern. Hence, though requiredfor making the very definition of real numbers sensible, the existence proof of nonempty domainsof magnitudes cannot be carried out in the theory FMR itself, and, a fortiori , in FMR ′ , in whichthat definition is stated.Two alternative strategies seem possible to deliver it. The first is in line with Frege’s per-spective and looks for an alternative way to prove the existence of continuously many objects onwhich continuously many permutations, forming a domain of magnitudes, can be defined. Thesecond departs from this perspective, and uses mathematical results unavailable to Frege. Itmight be appealed to, as a sort of unhoped lifeline for Frege’s purpose, in order to avoid someproblems the former strategy suffers from. It consists of inquiring whether continuously manypermutations forming such a domain can be obtained from countably many objects, whose ex-istence might, if needed, be proved by applying the previous proof-pattern. Let us call the firststrategy ‘inflationary’ and the second ‘non-inflationary’. The inflationary strategy can be implemented in at least two slightly different ways, in our setting.One follows Frege’s own plan sketched in § II.164, and takes the existence of natural numbersfor granted. The other appeals, instead, to a restricted version of BLV, to get an ω -sequence ofobjects other than Frege’s natural numbers. The structural similarity of these approaches makesthem suffer from the same difficulties. We merely consider the former. The reader might get anidea of the latter from the way we deal with a restricted version of BLV at the beginning of § < n, { m i } ∞ i =0 > composed by a natural number and an infinite sequence of positive natural numbers. These pairsare apt to code Cauchy series like(6.1) n + ∞ X j =1 λ j j ( λ j ∈ { , } ) , under the condition that m i is the i -th value of j such that λ j = 1 and the λ j are not all 0 aftera certain range. It follows that, once addition is appropriately defined on these pairs, one can We shall hark back on this matter in § α , a binary (first-level) relation R α such that, for any pairof these same pairs x and y , x R α y if and only if x + α = y . It is easy to see that this allows todefine as many relations as pairs like < n, { m i } ∞ i =0 > , namely continuously many ones, and thatthese relations are such that:– both they and their inverses are functional, since, for any such pairs x , y and z , x + α = y ∧ x + α = z and y + α = x ∧ z + α = x each entails that y = z ;– their composition mimics an addition on the pairs they are defined on, since, if β is alsosuch a pair, R α ⊔ R β extensionally coincides with R α + β , which is proved by observing that,for any such pairs x and y , x + ( α + β ) = y if and only if there is such a pair z such that x + α = z ∧ z + β = y ;– the identity relation is not one of them, since no Cauchy series like (6.1) is equal to zero.It is, then, easy to verify that these relations form a positive class, from which a domain ofmagnitudes is obtained by merely adding their inverses and the identity relation.Objects rendering these pairs in a formal setting are quite easy to define in any second-orderversion of arithmetic. A simple way to do it (Panza, 2016 and Panza, FC3) is by adding a newaxiom, under the form of the following abstraction principle:(FP) ∀ N XY ∀ N xy [ < x, X > = < y, Y > ⇔ ( x = y ∧ ∀ z ( Xz ⇔ Y z ))] , where the index ‘ N ’ signals that the universal quantifiers are restricted to properties of naturalnumbers and to these very numbers respectively—the acronym ‘FP’ stands fro ‘Frege’s Principle’,and emphasizes the fact that this principle is a restricted adapted form of BLV.Of course, to go ahead, we have to prove that FP is consistent. Assuming the consistency ofsecond-order arithmetic, to this purpose, it is, however, enough to observe that FP has a modelin the V ω +1 stage of ZF’s hierarchy. This is because second-order arithmetic has a model in theV ω segment of ZF, and consequently the set ℘ω of all subsets of the set of natural numbers is atstage V ω +1 , and provides the required model.This having been established, we can look at the pairs like < n, P > , implicitly defined byFP, and formed by a natural number n and a property P of natural numbers, and select amongthem those whose second element P is an infinite property of natural numbers—i.e. it is suchthat ∀ N x [ P x ⇒ ∃ N y [ x < y ∧ P y ]]. For short, call them ‘bicimal pairs’. Clearly, FP allows todistinguish continuously many such pairs.They can be arranged into two partitions, such that any bicimal pair < n, P > belongs toone partition if P
0, and to the other if ¬ P
0. A total order can, moreover, be defined on thesepairs, in such a way that the pairs in the former partition count as positive, the pair < , N + > (where ‘ N + ’ designates the property of being a positive natural number) counts as the zero pair,and the other pairs in the latter partition count as negative (more details are given in Panza,2016, p. 417; others will be found in Panza, FC3). This makes the bicimal pairs form an additiveAbelian group, that can be proved to be dense, totally ordered and Dedekind-complete (and, then,Archimedean), and can also be extended to a field by an appropriate definition of multiplication(details are, again, to be found in Panza, FC3). It would, then, be not only very tempting, butalso rather natural to code the real numbers by bicimal pairs, so as to avoid the very definitionof domains of magnitudes and of ratios thereof as perfectly useless.24till, this is certainly not what Frege’s strategy should lead us to. In order to follow hisindications, one should rather define appropriate permutations on bicimal pairs and show thatthey form a domain of magnitudes. This can easily be done by associating to any such pair < n, P > the binary relation R
New V , i.e. ∀ F ∀ G ( εF = εG ↔ ( Small ( F ) ∨ Small ( G ) → F ≡ G )), where ‘ Small ’ means ‘not equinumerous with the universal concept [ x : x = x ]’.By remaining faithful to this suggestion, we should replace ‘ Small ’ by ‘Fin’ (or take the former to mean ‘finite’),and FinBLV by ‘ ∀ F ∀ G ( εF = εG ↔ ( F ( F ) ∨ F ( G ) → F ≡ G ))’. Though this latter principle would not beequivalent to FinBLV, we cannot see any relevant difference between them with respect to our purpose. We preferFinBLV simply because it makes immediately clear that infinite concepts do not matter, here, by ascribing to themno extension, rather than ascribing to all of them the same extension. See footnote (34), above. The proof depends on the lemma that ∀ x ∃ y [ y = ε [ z : z = x ]]. Here is how this can be proved. FinBLV and ∀ X ∀ x [ Xx ⇔ Xx ] imply that ∀ ( F ) X [ εX = εX ], and, then, ∀ X ( F ) ∃ y [ y = εX ]. What is required is, then, to provethat ‘[ z : z = x ]’ denotes a (finite) property, which is ensured by predicative comprehension with parameters, sinceit entails that ∀ x ∃ X ∀ z [ Xz ⇔ z = x ]]. Notice that this proof also holds in (negative) free logic: thanks to LudovicaConti for drawing our attention to this; see also Conti (2019, p. 145, fn. 426, and pp. 151-152). This lemma im-plies, a fortiori , that ∀ E ⊘ x ∃ y [ y = ε [ z : z = x ]]. The very definition of the weak ancestral of S allows, then, to provequite easily the principle of induction for the FinBLV- abstracta having the property E ⊘ (or E ⊘ - abstracta , from nowon)—namely ‘ ∀ X (cid:20)(cid:18) X ⊘ ∧ X S −→ E ⊘ X (cid:19) ⇒ ∀ E ⊘ x [ Xx ] (cid:21) ’, where ‘ X S −→ E ⊘ X ’ abbreviates ‘ ∀ E ⊘ x ∀ y [( Xx ∧ x S y ) ⇒ Xy ]’.With this principle at hand, it is, then, equally easy to prove that ∀ E ⊘ x ∃ X [ x = εX ], as a consequence of thestipulation that ⊘ = ε [ x : x = x ], from which it immediately follows that ∃ X [ ⊘ = εX ]. By appealing to ∀ E ⊘ x ∃ y [ y = ε [ z : z = x ]], one gets that ∀ E ⊘ x ∃ y [ ∃ X [ x = εX ] ∧ y = ε [ z : z = x ]], that is, ∀ E ⊘ x ∃ y [ x S y ]. Next, theproperties of the ancestral of a binary relation allow to prove that ∀ E ⊘ x ∀ y [ x S y ⇒ E ⊘ y ], and so to conclude that ∀ E ⊘ x ∃ E ⊘ y [ x S y ]. The existence of countably many E ⊘ - abstracta finally follows from proving by induction that ∀ E ⊘ x [ ¬ x S ∗6 = x ], where S ∗6 = is the strong ancestral of S . This last part of the proof is a little bit harder than theprevious ones, but requires no specific skills: it just parallels the analogous proof of FA, by exploiting, like this(together with the principle of induction and some properties of the strong ancestral, also) the obvious facts that ¬ ⊘ S ∗6 = ⊘ and ∀ x, y, z [( x S y ∧ z S y ) ⇒ x = z ]. Though quite simple, this proof of existence of countably many E ⊘ - abstracta directly involves most of Peano’s second-order axioms as theorems about them. The axioms that donot enter it, i.e. ‘ ¬∃ E ⊘ x [0 S x ]’ and ‘ ∀ x, y, z [( x S y ∧ x S z ) ⇒ y = z ]’ (the second of which would allow replacing therelation S by the successor function), are, moreover, even easier to prove. Hence, if impredicative comprehensionis accepted, Peano’s second-order arithmetic is interpretable on the mentioned extensions—with no appeal to set-theoretical assumptions on them. These extensions do not comply, however, with HP, which makes them cruciallydiffer from natural numbers as defined in FA. , which prevents both from defining on Σ N a total order compatible with thegroup structure, and from making an injective map from a (nonempty) domain of magnitudes (ifany) into it surjective. It follows that Σ N does not provide a nonempty model for our Fregeanconsistent definition of such a domain. Nevertheless, by a theorem by Karrass and Solitar (1956,p. 65), Σ N provably “contains a copy of the additive group of the reals”. In other terms, thereis a subgroup of Σ N which is isomorphic to ( R , +), and is, then, not only totally ordered, butalso Abelian, dense, Archimedean and Dedekind-complete. Since any totally ordered, dense andDedekind-complete group of permutations is a model of our definition of domains of magnitudes , Σ N contains such a model. Insofar as Σ N is isomorphic to the symmetric group on whatever infinitecountable set, it is also so to the symmetric group Σ E ⊘ formed by the E ⊘ - abstracta . Hence, Karrasand Solitar’s theorem entails that admitting the existence of this latter symmetric group ensuresthe existence of a nonempty domain of magnitudes. It would, then, be enough to admit that itmakes sense to speak of all permutations on an infinite countable domain D of objects , and thatthe existence of (the elements of) D ipso facto entail the existence of the group formed by thesepermutations, to conclude that the existence of a nonempty domain of magnitudes is ensured bythe existence of the natural numbers or of the E ⊘ - abstracta , because of an immediate corollary ofKarrass and Solitar’s result .Karrass and Solitar’s proof could certainly not have been within Frege’s reach. But it isnot very entangled, as such, and, under the mentioned admission, most of it can be conductedconstructively, which is something Frege seems to require for his proof.Let I be an infinite countable set, for example an infinite subset of N . Let σ = ` i ∈ I C i be apartition of N in infinite (countable) subsets C i ( i ∈ I ), which makes S i ∈ I C i = N and C h ∩ C k = ∅ ,for any distinct h, k in I . Let ̺ : N −→ I be the (surjective) application defined by this partition,associating to any n in N , the single element i = ̺ ( n ) of I such that n ∈ C i . Such a partition, andtherefore such an application, can easily be defined constructively. To make a simple example,take I to be the set of all prime numbers plus 0, C the set of all natural numbers that are not(positive) powers of a prime number, namely { , , , , . . . } , and, for any prime number p , C p the set of all (positive) powers of p . Though we would not be (presently) able to write a generalformula providing, for any natural number n , the value ̺ ( n ) of i , such that n ∈ S ̺ ( n ) , there is afinite algorithm allowing us to calculate such a value ̺ ( n ) for whatever given natural number n .For any i in I , let π i be a permutation on C i . Define π : N −→ N by establishing thatfor any n in N , π ( n ) = π ̺ ( n ) ( n ). This is clearly a permutation on N , so that π ∈ Σ N . If ` i ∈ I S i and ̺ have been defined constructively, any π i , and therefore π , can also be so defined.Supposing ̺ be defined as in the previous example, we might, for example, take π i to be thepermutation exchanging the (2 j − C i , according to the usual order on N , with the 2 j -th one, and vice versa ( j = 1 , , . . . ). Then π would permute any natural numberwith another natural number following or preceding it in this subset, according to this order: A torsion element of a group G is an element g of G such that g n = e for some natural number n , where e isthe identity element of G . More precisely, the second-order property of being a permutation belonging to any such group satisfies theright-hand side of (3.6 ′ ). As Frege himself should have admitted, in order to make his own definition of domains of magnitudes sensible. Notice that though the definition of S ∗ = allows proving that the E ⊘ - abstracta form an ω -sequence, as showedin footnote (39) above, all that is relevant here is the cardinality of set formed by these abstracta, namely the factthat this set is countably infinite. (0) = 1, π (1) = 0, π (2) = 4, . . . Insofar as the same can be done for any permutation π i onany C i , the application π : N −→ N defined by stating that π ( n ) = π ̺ ( n ) ( n ) provides a groupmonomorphism Q i ∈ I Σ C i −→ Σ N , where Q i ∈ I Σ C i is the product of the symmetric groups on allsets C i —since, if { π i } i ∈ I and { π ′ i } i ∈ I are two families of permutations on all the sets C i , then { π i } i ∈ I ◦ { π ′ i } i ∈ I = { π i ◦ π ′ i } i ∈ I . Under the condition mentioned above, and provided all sets C i be (constructively) given, this further step of the proof is, also, constructively licensed.By Cayley’s theorem, any group G is isomorphic to a subgroup of the symmetric group Σ G on G itself. Thus, there is a group monomorphism ( Q , +) −→ Σ Q from the additive group ofthe rational numbers ( Q , +) into the symmetric group Σ Q on Q . Though quite general, Cayley’stheorem can easily be proved constructively, which also makes this new step of Karrass andSolitar’s proof admissible from Frege’s perspective. A quite usual way to prove it is, for example,by considering the translation τ y : x y ∗ x on the domain of G (where ∗ is the compositionlaw of this group), since it is easy to see that τ ( a ∗ b ) = τ a ◦ τ b , for any a , b in such a domain. Thisproof directly applies to the present case, for G is nothing but ( Q , +). To this purpose, we cantake Q to play the role of the domain of G and + that of ∗ , and observe that τ ( q + r ) = τ q ◦ τ r ,for any q , r in Q . Notice, moreover, that this application is perfectly akin to Frege’s suggesteddefinition of permutations on the pairs < n, { m i } ∞ i =0 > in the outline of his existence proof. For any i in I , let us choose a bijection ϑ i : Q −→ C i from the set Q to the set C i . Because ofthe monomorphism ( Q , +) −→ Σ Q , this engenders, for any such i , a new group monomorphism( Q , +) −→ Σ C i from ( Q , +) into the symmetric group Σ C i on any C i , and, by composition,a further group monomorphism Q i ∈ I ( Q , +) −→ Q i ∈ I Σ C i from the product Q i ∈ I ( Q , +) of countablymany copies of the additive group ( Q , +) into the product Q i ∈ I Σ C i . Again, if the partition π andthe application ̺ are defined constructively, making the choice of the bijections ϑ i and so gettingthese two monomorphisms require no form of the Axiom of Choice, and, under the conditionmentioned above, is unquestionably constructive. By combining the monomorphisms Q i ∈ I Σ C i −→ Σ N and Q i ∈ I ( Q , +) −→ Q i ∈ I Σ C i , we finally get, constructively again, a final monomorphism(6.2) Y i ∈ I ( Q , +) −→ Σ N . This provides a constructive ground for Karrass and Solitar’s proof. But, for its completion,a last step is needed, which, instead, requires non-constructive means and makes, then, thewhole proof non-constructive. Both additive groups ( Q , +) and ( R , +), enriched with the usualmultiplication by a rational number, can be regarded as vector spaces over the field ( Q , + , · ), andthis is also the case for the product Q i ∈ I ( Q , +). When Q i ∈ I ( Q , +) and ( R , +) are so regarded, it ishowever not plain that they have a basis, unless Zorn’s lemma is appealed to, since this lemmaallows to prove that every vectorial space has a basis . The non-constructive step of the proof A group monomorphism is an injective group homomorphism, i.e. an injective map µ from a group ( G , ∗ ) toanother group ( H , ⋆ ), such that µ ( x ∗ y ) = µ ( x ) ⋆ µ ( x ) for any x , y in G . See § By speaking of basis of a vector space, we more precisely mean, here, a Hamel basis. Let V be a vector spaceon a field F . An Hamel basis of V is a set B V of linearly independent vectors in V , such that for any vector v in V there is a (unique) finite subset { v , v , . . . v k } of B V such that v = a v + a v + . . . + a k v k , where a , a ,. . . , a k are elements of F . Q i ∈ I ( Q , +) and ( R , +) have a basis.This makes it possible to appeal to a new theorem ensuring that, if a vector space has severaldistinct bases, all of them have the same cardinality—which is, then, to be regarded as thedimension of this space. For vector spaces with finite bases, this is quite easy to prove . For vectorspaces whose generating sets are infinite, as Q i ∈ I ( Q , +) and ( R , +), the proof is more entangled, butcan still be given constructively. In this case, the theorem can indeed be viewed as an immediatecorollary of another theorem asserting that the cardinality of any generating set of a vector space V that can be regarded as the direct sum of an infinite family { V λ ∈ Λ } of non-zero vectorialsub-spaces is greater or equal to the cardinality of the set of indices Λ (Bourbaki, Algebra I, ch.II, prop. 23, cor. 2).Provided that two vector spaces on the same field (both having bases) are isomorphic if (andonly if) they have the same dimension, Q i ∈ I ( Q , +) and ( R , +), when regarded as vector spacesover ( Q , + , · ), have the same dimension, and thus are isomorphic. This obviously entails that thegroup monomorphism (6.2) results in a new group monomorphism( R , +) −→ Σ N , which makes Σ N contain a copy of ( R , +), as was required to be proved.If this proof is granted, together with the existence both of an infinite countable set D —as N or the set formed by the E ⊘ - abstracta —and of the symmetric group Σ D on this set, the conclusionfollows that there is (at least) a totally ordered, dense and Dedekind-complete subgroup of Σ D .Let ( M D , ◦ , < ) be such a subgroup of Σ D . Claiming that ( M D , ◦ , < ) is a domain of magnitudesin agreement with definition (3.6 ′ ) is the same as claiming that the triple ( M D , ⊔ , P D ) satisfiesthe open formula ‘[ P ( Y ) ∧ ∀ R [ X R ⇔ ð ( Y ) ( R )]]’ entering the right-hand side of this definition,with ‘ M D ’ as a value of ‘ X ’ and ‘ P D ’ as a value of ‘ Y ’, provided that: any binary relation R hasthe property M D if and only if it is in M D , and the property P D only if it has the property M D ;‘ ⊔ ’ denotes the same operation on the binary relations that are in M D as that denoted by ‘ ◦ ’; and,for any R, S in M D , P D R ⊔ S − if and only if S < R , so that P D R if and only if 0 M D < R —where0 M D is, of course, the neutral element of ( M D , ◦ , < ), namely the identity relation. To make thisclaim sensible, we have, of course, to grant that these conditions ensure the existence of the twosecond-level properties M D and P D , which requires third-order comprehension. Supposing itis admitted, Karrass and Solitar’s result provably establishes that, under the admission of theexistence of D and Σ D , there is a nonempty domain of magnitudes, namely the ordered groupdefined by the triple ( M D , ⊔ , P D ), since it entails that the properties P D and M D are such that P ( P D ) and ∀ R [ M D R ⇔ ð ( P D ) ( R )], so that M ( M D ).This being granted, it is, then, easy to prove that there are as many distinct binary relationsin such a domain as distinct ratios defined on it according to EP ∗ , namely that these ratios arecontinuously many. In other terms, the ordered pairs [ R, S ] of binary relations, the first of whichhas M D and the second P D , form continuously many distinct equivalence classes according tothe equivalence relation on the right-hand side of EP ∗ , under the replacement of both ‘ X ’ and A simple combinatorial argument allows to prove that the cardinality of any set of linearly independent vectorsis smaller or equal to the cardinality of whatever generating set. Insofar as a basis of a vector space is a set oflinearly independent vectors that generates the whole space, from this it immediately follows that two bases cannothave different cardinality, since, if they did, there would be a set of linearly independent vectors whose cardinalityis greater than that of a generating set. Y ’ with ‘ P D ’ . For shortness and simplicity, let us sketch this proof with reference to EP. Itsrephrasing with respect to EP ∗ is only a matter of laborious routine.Consider the first of the three disjoints forming the right-hand side of EP. Suppose that therebe three binary relations R , S , and S ′ that have P D and are such that S ❁ P D S ′ , ∀ N x, y [ x R ❁ P D y S ⇔ x R ❁ P D y S ′ ] . If there were, then, another binary relation T such that S ′ coincided with S ⊔ T , it would followthat ∀ N x, y [ x R ❁ P D y S ⇔ x R ❁ P D y ( S ⊔ T )] , which is impossible because of the Archimedeanicity of ( M D , ◦ , < ). This proves that, for whateverbinary relation R that has P D , there are as many distinct ratios R [ R , S ] (where S is a binaryrelation that has P D ) as binary relations that have P D , namely continuously many such ratios.Consider a binary relation R ′ that has P D and is distinct from R . Because of the completenessof positive classes, there is one and only one binary relation S that also has P D such that ∀ N x, y [ x R ❁ P D y S ⇔ x R ′ ❁ P D yS ] , which shows that there as many pairs of binary relations R and S that have P D , such that theratio R [ R, S ] is the same as R [ R , S ], as binary relations that have P D , and there is no pair ofbinary relations R ′ and S ′ that have P D such that the ratio R [ R ′ , S ′ ] is distinct from all ratios R [ R , S ], where S is a binary relation that has P D .Insofar as an analogous argument also applies, mutatis mutandis , to the second of the threedisjoints forming the right-hand side of EP, and the third of these disjoints only concerns orderedpairs of binary relations whose first element is the identity relation and makes all ratios whosedenominator is such relation identical, this is enough to conclude that the cardinality of the set { R [ R, S ] : M D R ∧ P D S } is the same as that of { R : M D R } , namely 2 ℵ , as was to be proved.Hence, defining real numbers as R - abstracta over a domain of magnitudes entails the existenceof at least 2 ℵ such numbers, since there is just one real number for any equivalence class inducedby the right-hand side of EP or EP ∗ on a single domain of magnitudes. To prove that thereare just ℵ , recall that, as observed in § ∗ it follows that, if therewere several such domains, for any R - abstractum on one of those domains, there would just beone R - abstractum on the others that is identical with it, which entails that, if there were severaldomains of magnitudes, the R - abstracta on one of them would be just the same objects as the R - abstracta on the other. Given all the previous considerations, we can finally tackle two major questions concerning thedefinition of reals in FMR or FMR ′ : i ) Is there a strong enough sense in which this definition islogical? ii ) Is this definition independent of natural numbers and their theory? Insofar as it seemsdifficult to imagine a consistent definition which is closest to Frege’s envisaged one, the answer tothese questions is relevant to assess Frege’s achievements as well: Was Frege’s plan for defining real Notice that from the conditions above, it immediately follows that ð ( P D ) ( R ) if and only if M D R . .All of them boil down to two issues: whether FMR or FMR ′ can be taken to be logicalsystems, independent of a previous definition of natural numbers (likely got through FA); whetheran existence proof of nonempty domains of magnitudes and of real numbers as defined in FMRor FMR ′ is compatible with the logicality and arithmeticity of these definitions. Insofar as FMR and FMR ′ are obtained by adding some new axioms to L PCA , we will beginby investigating whether this latter system is genuinely logical and independent from the naturalnumbers. Both issues also apply to our definition of domains of magnitudes within it.Likely, no one would question its independence from the natural numbers. The considerationsadvanced in § . Still, admitting that L PCA indeedhas these features is not enough for concluding that our definition of domains of magnitudes is, inturn, independent of natural numbers and logical in some more significant sense than the simpleand quite weak one of being formulated within a logical system. There are two concerns, here.The first is that, even in a logical system, it seems possible to define items whose logicalnature is suspect. Panza (2018) and Panza (FC2) already raised the question in relation bothto natural numbers and magnitudes, as originally defined by Frege as appropriate extensions.Surely, according to our reformulation of Frege’s definition, magnitudes are no more extensions,but rather binary first-level relations. Still, apart from the identity relation, the relations formingsuch domains are not identified as particular relations somehow precisely defined; they are rathercharacterized as possible places in whatever system exemplifying a certain structure. This makesthis definition define domains of magnitudes, but not magnitudes as such, which is perfectly inline with Frege’s remark quoted in §
2. Hence, all that the definition fixes is the structure of adomain of magnitudes, not its content, which is to be given independently of it.As such, this might even be taken as an argument for its logicality, if, contra
Frege, it isadmitted that logic has no content. But it makes the second concern crucial. As alreadyclaimed, whereas an existence proof of nonempty domains of magnitudes cannot be providedwithin L PCA (as well as within FMR and, a fortiori , FMR ′ ), it is indispensable for makingour definition of real numbers sensible. So, proving, necessarily outside these systems, the ex-istence of a nonempty domain of magnitudes is an essential part of this very definition (even ifthis is not required to formulate the definition of domains of magnitudes themselves), not onlyof a model-theoretical enquiry on it. This makes both the logicality and the arithmeticity of the Simons (1987, §
7) has stressed the crucial differences between Frege’s logicism for natural numbers and hisviews on real ones. Without undermining his arguments—which take however for granted the usual reading ofFrege’s logicism for natural numbers, which we rather take as questionable under many respects: See Panza (2018)and Panza (FC2)—we follow another strategy here: we frontally attack the idea that Frege’s envisaged definitionof real numbers might be taken as logical in any substantial sense. But see footnote (5), on this matter. ′ , neededfor conducting such an existence proof. Two major issues arise.The first is that it seems plausible to require that a genuinely logical definition not be inneed of an external existence proof of the items it defines—or of other associated ones. Sincesuch a definition should purportedly ensure the existence of these items by merely showing that,if there were none, some logical, or innocent enough, truths would not be true after all. Thisis indeed allegedly the case of the neologicist existence proof of natural numbers —which wemimicked in the existence proof of the E ⊘ - abstracta , in § ′ , nor that of domains ofmagnitudes in L PCA , can pretend to be logical in the same sense in which neologicists claimtheir definition of natural numbers is .This leads to the second issue: once admitted that the neologicist’s proof-pattern does notapply, and that this prevents our Fregean definitions of domains of magnitudes and real numbersto be logical in the above demanding sense, the question arises whether these definitions mightnevertheless be deemed logical in some less demanding sense . The question seems to havedifferent answers according to whether it concerns the former definition or the latter, and whetherdomains of magnitudes are regarded as such or as tools for defining real numbers. If we look atthe definition of domains of magnitudes as such, and admit that L PCA be a genuine logicalsystem, it is hard to find any other reason than that raised above to deny its logicality. But if welook at domains of magnitudes as tools for defining real numbers, the situation changes. Insofaras proving the existence of nonempty such domains is essential for enabling them to play thisrole, the question becomes whether the proof can be so shaped as to make it logical, and, then,part of a logical definition of these numbers. This is, then, the question we have to tackle, now.Above we explored two different strategies for conducting this proof: an inflationary and anon-inflationary one. In what follows, we will expand on them by considering how they score withrespect to the issue of logicality. Insofar as the question of the logicality of our definition of realnumbers has multiple interconnections with that of its arithmeticity, we will also consider in the To see it, consider the argument proving the existence of 0 that we have detailed at the beginning of § x : x = x ] exists not only by predicative, but also by logical comprehension, as well as logic is enoughto get that [ x : x = x ] ≈ [ x : x = x ], and HP is so for getting that 0 = 0, which could not be true if 0 did not exist.The same pattern allows to prove the existence of each natural number, provided that comprehension be extendedto formulas involving the operator ‘ x : x = 0] ≈ [ x : x = 0]’, which follows, in turn, by logic,from the existence of the concept [ x : x = 0], ensured by comprehension applied to the formula ‘ x = x : x = x ]’.On the other side, the existence of the totality of numeral numbers is proved by proving, by HP and impredictaivecomprehension, the successor axioms, which would be false, if these numbers did not exist. In fact, neologicists usually take their definition of natural numbers to be analytic, though not logical. Still,we made clear from the very beginning why we do not endorse this distinction here—see § We leave here apart the question of whether the neologicist definition of natural numbers or our definitionof the E ⊘ - abstracta are actually logical or analytical. In Panza (2016), pp. 420-423, the point is made that theformer definition might be deemed so in a quite peculiar sense, quite different than those that are current in thediscussion on logicism and neologism, and because of a completely different argument than the neologicist’s. Thereis no need to come back on this point, here. It is only important to observe that it does in no way apply to ourdefinition of domains of magnitudes and real numbers. A proof following the inflationary strategy may be deemed non-logical just because of its infla-tionary nature. The reason is obvious: insofar as no proof by countable induction is possible here,such a proof cannot but grant that the abstraction principle introducing the continuously manyobjects it concerns (FP, in our case) eo ipso entails the existence of these objects, which appearsto be incompatible with its being logical, and, then, part of a logical proof.One could object that the argument merely points out that the proof is not logical, becauseit requires means other than countable induction to prove the existence of continuously manyobjects, which is unfair, at best. After all, real numbers must be continuously many, so thataccepting this argument would amount to principledly excluding the possibility of a logical def-inition of real numbers ensuring their existence. The objection is not convincing. It is entirelypossible that no such definition be logical. Still, if a definition of these numbers is offered withthe aim of being so, it should, at least, avoid requiring an existence proof for continuously manyobjects other than the reals. This would leave room for arguing that proving the existence ofcontinuously many objects is not part of its job, but should be left for further meta-theoreticalconsiderations. The point is, then, that the inflationary strategy is not suitable for entering alogical definition of real numbers because it requires an existence proof of continuously manyobjects other than the reals: a proof which cannot but appeal to independent resources fromthose involved in the definition itself.Another essential feature of the inflationary strategy is also relevant for the present discussion:its delivering an arithmetical copy of the additive group of the reals as a condition for makingtheir definition appropriate. This makes clear both its arithmetical nature, and its essentialmathematical circularity. Let us consider the two allegations in turn.To reply to the arithmeticity allegation, it is not enough to argue that proving the existenceof a nonempty domain of magnitudes arithmetically does not make a definition of real numbersas ratios of magnitudes arithmetical. The fact that an existence proof of such a domain is anindispensable part of the definition immediately entails, indeed, that this definition can be deemednon-arithmetical only in the presence of an existence proof of a non-arithmetical nonempty domainof magnitudes. Since, if one could only prove the existence of arithmetical such domains, definingreal numbers as ratios on them would make them arithmetical items, after all . The point might be softened by observing that our Fregean definition of domains of magnitudes differs fromother possibly arithmetical ones for not appealing to any specific property of the objects on which the relevantbinary relations are defined. To better see this, we can compare this definition to one mimicking Dedekind’sdefinition by cuts in terms of binary relations (we thank Andrew Moshier for his suggestion). Let ( O , < O ) be atotally ordered set without endpoints, whose elements count as objects. By adopting a third-order logical systemwith third-order predicative comprehension, the following explicit definition can be provided, where the index ‘ O ’restricts the quantifiers to binary O -relations (i.e. binary relations among the elements of O ) and to these veryelements, respectively: ∀ O R C O R ⇔ ∀ O x, y [ xRy ⇒ x < O y ] ∧∀ O x, y, z, w [( xRy ∧ z < O x ∧ y < O w ) ⇒ zRw ] ∧∀ O x, y [ xRy ⇒ ∃ O z, w [ zRw ∧ x < O z ∧ w < O y ]] ∧∀ O x, y [ x < O y ⇒ ( ∃ O z [ xRz ] ∨ ∃ O w [ wRy ])] ∧∀ O x, y, z, w [( xRy ∧ zRw ) ⇒ xRw ] .
33o reply to the circularity allegation, one should argue that the copy of the additive groupof the reals is just a copy, since, though structurally coincident with real numbers, its elementsintrinsically differ from them. We can imagine Frege advancing this argument. But we can hardlyfollow him in this without making any working mathematician sarcastically smile.Let us recap. The inflationary strategy suffers from two problems: in absence of a furtherexistence proof of non-arithmetical domains of magnitudes, it makes real numbers themselvesarithmetical objects, after all; it requires a preliminary structural definition of the real numbers,in order to make the planned definition of these same numbers suitable . At least four reasons might be advanced to argue that, in the light of the existence proof in § i ) thatproof is based on the symmetric group Σ N on the natural numbers; ii ) it allows to conclude thata nonempty domain of magnitudes exists only if it is admitted that the symmetric group on aninfinite countable set exists if this set exists; iii ) it essentially appeals to the additive group ( R , +)of the real numbers themselves; iv ) it appeals to Zorn’s lemma, and is, then, not constructive.The first reason cannot be dismissed by merely observing that, in our reconstruction of theproof, Σ N has been replaced by the symmetric group on the set of E ⊘ - abstracta . Since, once Σ N is The third conjunct of the right-hand side entails that no binary O -relation has the property C O if ( O , < O ) is notdense. Let us suppose that it be so. Call a relation ‘ C O -relation’ if it is a binary relation having C O . We cansay that any C O -relation defines a cut on ( O , < O ). The collection of the C O -relations does not form a domain ofmagnitudes, in the sense established above, since the C O -relations are not permutations. Still, we might weakenFrege’s requirement on domains of magnitudes, and take such domains as constituted by totally ordered, denseand Dedekind-complete groups of first-level binary relations, independently of their being permutations. The C O -relations might, then, form a domain of magnitudes if a commutative (and associative) addition admitting aneutral element be defined on them. To this purpose, let us suppose that an addition + O be defined on ( O , < O ),so as to make ( O , < O , + O ) a totally ordered, Abelian and dense additive group. We can easily define an addition+ C O on the C O -relations, by stating that ∀ C O R, S ∀ O x, y (cid:2) x (cid:0) R + C O S (cid:1) y ⇔ ∀ O z, w, v, u [( zRv ∧ wSu ) ⇒ ( x < O z + w ∧ v + u < O y )] (cid:3) , where the index ‘ C O ’ to the first universal quantifier restricts it to these relations. The C O -relation Z O defined by ∀ O x, y [ x Z O y ⇔ x < O O < O y ]is the neutral element of + C O , and another C O -relation R , is deemed positive if and only if ∃ O x, y [ xRy ∧ O < O x < O y ] , and negative otherwise. One could, then, define an order relation ❁ C O on the C O -relations by stating that ∀ C O R, S (cid:20) R ❁ C O S ⇔ ∃ C + O T (cid:2) R + C O T = S (cid:3)(cid:21) , where the index ‘ C + O ’ restricts the existential quantifier to positive C O -relations. These would form a totallyordered, dense and Dedekind-complete group under + C O and ❁ C O , and, then, a domain of magnitudes, in theprevious weakened sense. One might, then, define real numbers as ratios on such a group. Still, so defined, realnumbers would be, structurally speaking, nothing more than ratios on cuts-relations on the additive group of therational numbers, and this would make them intrinsically arithmetical items, in a much stronger sense than thereal numbers defined in FMR or FMR ′ , under the condition that the only nonempty domains of magnitudes whoseexistence can be proved were arithmetical. Both problems also arise if the inflationary strategy is implemented by E ⊘ - abstracta . As for the first, thisis obvious. As for the second, notice that these abstracta could enter the existence proof only because of theirfeatures that make them structurally coincide with the natural numbers. N itself. That Σ N is isomorphic with the symmetric group over any infinite countableset is, indeed, simply because any such set can be put into a bijection with N , so that its elementscan be taken either to count as natural numbers or, at least, to be encoded by them.The second reason is similar to one discussed above as for the inflationary strategy: on whatlogical ground can we argue that the existence of a countable infinite set entails the existence ofthe symmetric group over this set—or, more in general, of an uncountable set somehow generatedby it by considering at once some totality of properties, relations or functions defined on theelements of this set? The fact that Frege himself suggests making a similar admission, in order toprove the existence of a nonempty domain of magnitudes, in no way makes it logically licensed.Rather, it seems to show that the very proof Frege suggested would have actually been not logical.The third and fourth reasons are by far more delicate, and somehow interconnected. Since, ifa constructive proof of Karrass and Solitar’s theorem were available, one could hope to rely onit in order to constructively define a totally ordered, dense and Dedekind-complete subgroup of Σ N , without recurring to ( R , +).To dismiss the third reason, and the circularity allegation that goes with it, one might replaceKarrass and Solitar’s theorem with a more general result not involving ( R , +). A natural candidateis a result by de Bruijn (1964, p. 594), according to which, for any infinite cardinal κ , every Abeliangroup of order 2 κ can be “embedded into” the symmetric group of a set of cardinality κ . Still,the basic idea of de Bruijn’s proof is not so different from Karrass and Solitar’s and makes thisproof also depend on the axiom of Choice, though avoiding appealing to vector spaces. The factthat the theorem does not specifically involve ( R , +) is, moreover, far from being an advantagein our perspective. Since it makes this theorem unable to provide a ground for the requiredexistence proof. For the purpose of this latter proof is establishing that Σ N (or, more generally,the group of permutations on a countable set) actually includes a subgroup complying with therelevant structure, while this theorem merely ensures that, if there is such a group, then it can beembedded into Σ N , and can be regarded as a subgroup of it. This makes, of course, de Bruijn’stheorem immediately entail that ( R , +) can be embedded into Σ N . This cannot but make thecircularity even more evident, since it is only the existence of R that can ensure that a subgroupof Σ N complying with the relevant structure exists. In order to solve the issue, one should provethat Σ N includes a totally ordered, dense and Dedekind-complete subgroup, without assumingthe existence of this group. To the best of our knowledge, this has not yet been done.This does not mean, of course, that this result, or any other entailing it, has not actually beenproved or, even less so, that this cannot be done. The fourth reason suggests, however, that therelevant question is not whether this has been or might be done, but, rather, whether this can bedone constructively, i.e. without appealing to a form of the Axiom of Choice, which might hardlybe taken as a logical principle. When put in a clear mathematical form, the question is whetherit is provable in ZF alone (or in some other appropriate setting that neither presupposes norentails the Axiom of Choice) that Σ N contains a totally ordered, dense and Dedekind-completesubgroup, and whether, moreover, this can be done without assuming the existence of ( R , +). Tothe best of our knowledge, this question also has not been answered yet. This theorem was firstly published one year later than Karrass and Solitar’s (de Bruijn 1957, pp. 560-61 and566), but it was then erroneously proved. The error lied with a lemma proved by erroneously supposing thata certain arbitrary group could be non-Abelian. The proof was later corrected and made independent of thislemma—and in fact simplified.
35o begin enquiring about it, one might wonder whether Karrass and Solitar’s proof can be freedfrom Zorn’s lemma or any equivalent assumption. Such an assumption enters the proof to ensurethat any vector space has a basis—i.e. that such a basis exists though it cannot be constructivelydisplayed. This makes it relevant to observe that Blass (1984) proves that the assumption thatany vector space has a basis is (ZF-)equivalent to the Axiom of Choice. This is still not enoughto ensure that the appeal to a form of the Axiom of Choice cannot be avoided in Karrass andSolitar’s proof, and that this proof is, then, both non-constructive and intrinsically dependent onsuch an axiom. Since what is required for this proof to work is not, properly, that any vectorspace has a basis, but rather that this is so for the two relevant such spaces, i.e. Q i ∈ I ( Q , +) and( R , +). The issue becomes, then, whether one can prove that these very vector spaces have a basiswithout appealing to a form of the Axiom of Choice or to any other non-constructive means. Tothe best of our knowledge, anew, this is still unknown.Still, even if this could not be done, it would not follow that Karrass and Solitar’s theoremcannot be proved without appealing to a form of this axiom. The only occurrence of Zorn’s lemmain the previous proof is, indeed, in its very last step, which is the only one involving vector spaces.It is, then, natural to wonder whether the theorem could be proved by avoiding this step (and,then, presumably any reference to vector spaces), by replacing it with another step not dependingon the Axiom of Choice or some other non-constructive assumptions.We could imagine two scenarios. In the first, the question is whether ZF alone is capable ofproving Karrass and Solitar’s theorem: to this extent, either it is, or the theorem is undecidablethere. In the second scenario, the question is whether ZF augmented with some axioms incom-patible with the Axiom of Choice, such as the Axiom of Determinateness, is capable of provingthis theorem. To the best of our knowledge, these issues have also not been settled yet .The conclusion to be drawn from all these remarks cannot be but prudent. Still, it can certainlyno more suggest that Karrass and Solitar’s theorem provide a basis to argue that our Fregeandefinition of real numbers is logical. Since, circularity issues aside, arguing that this theoremcan enter a non-inflationary existence proof for a nonempty domain of magnitudes, suitable formaking our definition logical, would be quite premature, at best. And it would be even more soto argue that a more general theorem, asserting that Σ N includes an appropriate group identifiedwithout appealing to R , can enter such a non-inflationary existence proof. The previous considerations suggest that there is no way to prove the existence of nonempty do-mains of magnitudes without wiping out both the logicality and non-arithmeticity of our Fregeandefinition of real numbers. In light of this conclusion, one might suggest changing the rules of thegame. Even if there is no way to prove, by (higher-order) logic, suitable existentially innocentabstraction principles and appropriate algebraic and/or set-theoretical constructive arguments,that nonempty domains of magnitudes exist, still we know they do. For we can show or prove it, First introduced by Mycielski & Steinhaus (1962), this axiom asserts that “certain infinite, deterministic 2-person games with complete information [. . . ] are determinate, i.e., that one of the players has a winning strategy”.See also Herrlich (2006, p. 151), which also provides a proof of the incompatibility between the Axiom of Choiceand the Axiom of Determinateness. There still might be clues on the second issue: while the Axiom of Choice entails that R , as a vector spaceover ( Q , + , · ), has bases, the Axiom of Determinateness entails that it has not (Herrlich 2006, Theorem 4.44 andCorollary 7.20)—and one might even guess that it be the same for an infinite product of copies of ( Q , +). ´a la Frege within FMR or FMR ′ , even with no existential proof, allows one toshow that ratios on any externally given domain of magnitudes, whether intrinsically arithmeticalor not, are real numbers. Since, as we have already seen, taking a real number to be a ratio ondistinct domains of magnitudes is nothing but describing the same object in different ways—orgiving different names to it.The problem with this move is that applying our definition to whatever externally given do-main of magnitudes would certainly warrant that the ratios on it are real numbers, but not thatreal numbers are intrinsically such ratios, let alone that they are non-arithmetical items. If wereasoned this way, we would do nothing essentially different from appealing to a representationtheorem to draw the conclusion that real numbers measure the magnitudes in the relevant do-mains, in the spirit of the measurement theory . In both cases, all we do is recognize that someexternally given systems (arithmetical or not) comply with some fixed structural conditions. Thefact that these structural conditions are fixed by our definition in FMR or FMR ′ , or by recurringto algebraic axioms as those of a totally-ordered complete Abelian field (as usually supposedin measurement theory), or, again, by alternative definitions (as Cantor’s and Dedekind’s, byCauchy’s sequence and cuts on rationals, respectively, or even as the one grounded on FP) makesno essential difference on this matter.In the eyes of a Frege partisan, there would be a crucial difference only if the existence proofwere deemed an essential, though supplementary, part of the definition itself, as we did above.Since this would make the numbers so defined intrinsically ratios on domains of magnitudes, andtheir application to measurement “built into” their nature and/or their very definition, as requiredby the application constraint (Wright, 2000, p. 325). In this respect, the previous remarks onthe arithmeticity and logicality of our definition in FMR or FMR ′ should be intended to suggestthat compliance with this constraint is incompatible not only with offering a logical definitionof real numbers, as already argued in Panza & Sereni (2019), but also with defining them non-arithmetically, despite Frege’s adhesion to the same constraint as the main source of his quest fora non-arithmetical definition of these numbers. Up to now, we have only considered the existence proof of nonempty domains of magnitudes.Still, the indispensability and the nature of this proof are not the only reasons suggesting that ourFregean definition of real numbers is neither logical nor non-arithmetical. Since, once domains ofmagnitudes have been defined and somehow proved to exist, the question remains open of definingreal numbers as ratios on them. In our setting, this is done by means of EP ∗ . The questions About the tension between considering applications of real numbers in agreement with the measurement theoryand taking them to be ratios on domains of magnitudes, see the Hale-Batitsky discussion in Hale (2000, 2002) andBatitsky (2002). On this matter see also Panza & Sereni (2019, pp. 122-123 and 126-130). Possibly with the help of (CA ), if an explicit definition like (5.2) is required. We do not want to enterhere a discussion about the logicality of (the different sorts of) comprehension. We merely remark that the highimpredicativity of (CA ) might make many doubt not only its logicality, but also its licitness. Who doubts bothhas no other choice but rejecting definition (5.2) and rest content with (5.2 ′ ). Who doubts only the former caneither admit definition (5.2), but take it as non-logical, or rejecting, again, this definition in favor of (5.2 ′ ). PCA , by involving a piece of informal language allowing for predicate variables like ‘ xR ’.The two features are connected, since the quantification over natural numbers just operates on theindividual variables occurring in these predicate variables. Replacing EP with EP ∗ allows avoidingboth the quantification over natural numbers and the appeal to informal language at once. Surely,EP ∗ involves no second-order predicate constant supposedly designating the property of being anatural number. Still, this does not ensure, yet, its independence from natural numbers, sinceit is far from clear that the trick used to avoid the quantification on these numbers is actuallyindependent of them. What might make one suspect it is not that the right-hand side of (5.1)is just an appropriate third-order rephrasing of the right-hand side of HP. Hence, if it wereadmitted that, when applied to finite concepts, HP is intrinsically inherent to natural numbers—not only because the objects not complying with it are not natural numbers, but also because itsassumption ipso facto brings these latter about—, one should infer that, in spite of appearance,also EP ∗ depends on these numbers. As a matter of fact, this is a strong assumption, but one thatcan be made in a Fregean vein, and which might bring, then, ipso facto —that is, independently ofany consideration on the existence proof of nonempty domain of magnitudes—, to the conclusionthat our definition of real numbers, whether in FMR or FMR ′ , is essentially arithmetical.Someone admitting this assumption might still argue against this conclusion by observing that(5.1) essentially differs from HP for being a (metalinguistic) abbreviation stipulation, rather thanan axiom providing an implicit definition of a functional constant. This is enough, one mightcontinue, to make EP ∗ appeal to no variable ranging on objects that might count as the naturalnumbers. This is unquestionably so. However, any instance of ‘ ( X , X ′ ) E ( R, T, R ′ , T ′ )’ asserts thata certain first-level binary relation is the same multiple of another such relation over a certainpositive class as a third such relation of a fourth one, over the same or another positive class.This is, in turn, the same as asserting that the iterations of the composition operation on sucha relation within the former class are into a bijection with the iterations of composition on sucha relation within the latter class. If this is not the same as making natural numbers enter intoplay, it is, at least, the same as making the equinumerosity relation so. Hence, if EP ∗ is notdependent on natural numbers, it seems to be, at least, dependent on counting. There is no easyway to settle whether this is enough to make EP ∗ an arithmetical principle. Here, we just observethat this makes our Fregean definition of real numbers, whether in FMR or FMR ′ , dependent onan essential ingredient of any Fregean definition of natural numbers. Even if this, as such, doesnot make our definition arithmetical, it is plausibly enough for making it much more related tonatural numbers than Frege might have desired his definition be.Let us come, now, to the first question: can EP ∗ be deemed logical? A simple way to tacklethe question might be that of choosing between two quite natural options: either any abstractionprinciple is logical if it is stated through a logical language, or it is so only because of the peculiarnature of its right-hand side. In the former case, EP ∗ is logical if L PCA is so. In the lattercase, EP ∗ cannot be logical on the same grounds on which HP or a consistent version of BLVmight be so. If this simple alternative is rejected, if only for argument’s sake, what criterionmight be provided to distinguish logical abstraction principles stated in a logical language fromnon-logical ones? Consistency is surely not enough. But, then, what? We cannot dwell on thisissue here. We simply contend that the burden of the proof seems to be on anyone arguing that38P ∗ is logical, despite its being essentially akin to Euclid’s definition of proportionality, which hasbeen considered for centuries as the cornerstone of the most fundamental mathematical theoryon which classical geometry was crucially grounded. Though some of them are certainly far from knock down ones, we think we advanced enougharguments in favor of the claim that our rendering of Frege’s envisaged definition of real numbersis neither logical nor non-arithmetical. As our rendering is arguably the closest possible to it, thisconclusion questions the possibility that Frege’s own definition could be achieved logically andnon-arithmetically. It remains to establish whether this was actually Frege’s intent.That Frege was aiming at a logical definition of real numbers as his main goal for the foundationof real analysis might be questioned for several reasons . One of them might be the following.From our reconstruction, it seems to emerge that arguing for the logicality of a definitionof real numbers following Frege’s indications requires arguing that FP, or any akin principle, isboth logical and capable of delivering continuously many objects without the assistance of anyindependent existence proof. But if so, then FP would also be enough for delivering real numbers,if not as logical objects, at least as objects defined in a logical setting. But, then, why did Fregeventure himself in a so entangled definition whose logical nature is as suspect as that of realnumbers as ratios of magnitudes?Possibly, far from considering logicality as his ultimate aim, he overall wanted to link realanalysis to a general theory of magnitudes. This has been argued for in Panza & Sereni (2019).Or, possibly, he merely wanted to distinguish real from natural numbers, making the formeressentially independent of the latter, for their being objects of an essentially different kind.Though the two possibilities are not incompatible with each other, our conclusion might betaken as a piece of evidence that he could not have reached this second aim by following theroute envisaged in the Grundgesetze . The first aim remains, which is certainly paramount froma purely mathematical perspective. If we admit that this was, after all, his prominent goal, thenour rendering of his definition might be taken as an indication of a simple way to accomplish it.
Acknowledgments:
We would like to thank Mirna Dˇzamonja, Alain Genestier, Paolo Mancosu,Andrew Moshier, Jamie Tappenden, an anonymous reviewer for
The Review of Symbolic Logic ,and the audiences of the conferences previous drafts of this paper were presented at, for usefulcomments and criticisms. We are also indebted to Mya MacRae for the linguistic revision of thepaper. The work of Marco Panza has been supported by the ANR-DFG project FFIUM.
References
Adeleke S.A., Dummett, M.A.E, Neumann, P.M. (1987) On a Question of Frege’s aboutRight-Ordered Groups,
Bull. Londin Maths. Scoc. , 19: 513–521.Antonelli, A. & May, R. (2005) Frege’s Other Program,
Notre Dame Journal of Formal Logic ,46(1): 1–17. Some are advanced in Panza (FC2).
Philosophia Mathematica , 10(3): 286–303.Blass, A. (1984) Existence of bases implies the axiom of choice, in J. E. Baumgartner, D. A.Martin and S. Shelah (eds.),
Axiomatic Set Theory , vol. 31 of
Contemporary Mathematics ,AMS: 31–33.B laszczyk, P. (2013) Nota o rozprawie Otto H¨oldera “Die Axiome der Quantit¨at und dieLehre vom Mass”,
Annales Universitatis Paedagogicae Cracoviensis. Studia ad DidacticamMathematicae Pertinentia , 5: 129–144.Boolos, G. (1998a) Iteration Again, in Boolos (1998c): 88–104.Boolos, G. (1998b) Saving Frege from Contradiction, in Boolos (1998c): 171–182.Boolos, G. (1998c)
Logic, Logic, and Logic , J. Burgess and R. Jeffrey (eds.), Cambridge, MA:Harvard University Press.Bourbaki, N. (Algebra I)
Elements of Mathematics, Algebra I , Springer, Berlin, Heidelberg,etc., without date.de Bruijn, N.G. (1957) Embedding Theorems for Infinite Groups,
Nederl. Akad. Wetensch.Proceedings , Ser. A, 60 (=
Indagationes Math. , 19(5)): 560–570.de Bruijn, N.G. (1964) Addendum to ‘Embedding Theorems for Infinite Groups’,
Nederl.Akad. Wetensch. Proceedings , Ser. A, 67 (=
Indagationes Math. , 26(5)): 594–595.Conti, L. (2019)
Paradossi di Russell e programmi astrazionisti. Spiegazioni e soluzioni aconfronto , PhD thesis.Dummett, M. (1991)
Frege: Philosophy of Mathematics , Duckworth, London.Euclid (EH)
The Thirteen books of Euclid’s Elements , translated with Introduction and Com-mentary by T. L. Heath, Cambridge University Press, Cambridge 1908 (3 vols.), Duckworth,London.Frege, G. (1884)
Grundlagen der Arithmetik , W. K¨obner, Breslau.Frege, G. (1893-1903)
Die Grundgesetze der Arithmetik , Verlag von H. Pohle, Jena, 1893-1903(2 vols).Frege, G. (1953)
The Foundations of Arithmetic (translated by J. L.Austin), Blackwell,Oxford, 1953.Frege, G. (2013)
Basic Laws of Arithmetic , translated by P. A. Ebert and M. Rossberg,Oxford University Press, Oxford, 2013.Gauss, C. F. (1831) Theoria residuorum biquadraticorum. Commentatio secunda,
G¨ottingis-che gelehrete Anzeigen , 1831, vol. I, n. 64, April, 23th, 1831, pp. 624-638.Gauss, C. F. (Werke)
Werke , Herausgeben von der K¨oniglichen Gesellshaft der Wissenshaftenzu G¨ottingen, 1863-1933, 12 vols. 40ale, B. (2000) Reals by Abstraction,
Philosophia Mathematica , 8: 100–123.Hale, B. (2002) Real Numbers, Quantities, and Measurement,
Philosophia Mathematica ,10(3): 304–323.Herrlich, H. (2006)
Axiom of Choice , Springer, Berlin-Heidelberg.H¨older, O. (1901) Die Axiome der Quantit¨at und die Lehre vom Mass.
Berichte ¨uberdie Verhandlungen der K¨oniglich S¨achsischen Gesellschaft der Wissenschaften zu Leipzig,mathematisch-physischen Classe , 53: 1–64.Karrass, A. & Solitar, D. (1956) Some remarks on the infinite symmetric groups,
Mathe-mathische Zeitschrift , 66: 64–69.Mycielski, J. & Steinhaus, H. (1962) A mathematical axiom contradicting the axiom ofchoice,
Bulletin de l’Acad´emie Polonaise des Sciences. S´erie des Sciences Math´ematiques,Astronomiques et Physiques , 10: 1–3.Panza, M. (2015) From Lagrange to Frege: Functions and Expressions, in H. Benis-Sinaceur,M. Panza & G. Sandu,
Functions and Generality of Logic. Reflections on Frege’s andDedekind’s Logicisms , Springer, Cham, Heidelberg, New York, Dordrecht, London: 59–95.Panza, M. (2016) Abstraction and Epistemic Economy, in S. Costreie (Ed.),
Early AnalyticPhilosophy. New Perspectives on the Tradition , Springer: 387–428.Panza, M. (2018) Was Frege a Logicist for Arithmetic?, in A. Coliva, P. Leonardi, S. Moruzzi(eds),
Eva Picardi on Language, Analysis and History , Palgrave MacMillan, Springer Nature,Cham, Switzerland: 87–112.Panza, M. (FC1)
Anzahlen et Werthverl¨aufe : quelques remarques sur les §§ I.34-40 des
Grundgesetze de Frege, forthcoming in
L’´epist´emologie du dedans. M´elanges en l’honneurd’Hourya Benis-Sinaceur , Garnier, Paris.Panza, M. (FC2) Was Frege a Logicist?, forthcoming (This is a much extended version ofPanza (2018), still unpublished).Panza, M. (FC3)
Reals by Abstraction. An enquiry about Epistemic Economy in Mathematics ,forthcoming monograph.Panza, M. & Sereni, A. (2019) Frege’s Constraint and the Nature of Frege’s FoundationalProgram,
The Review of Symbolic Logic , 12(1): 97–143.Payne, J. (2013) Abstraction Relations Need not be Reflexive,
Thought: A Journal of Phi-losophy , 2(2): 137–147.Schirn, M. (2013) Frege’s approach to the foundations of analysis (1874-1903),
History andPhilosophy of Logic , 34(3): 266–292.Scott, D. (1958-59)
A General Theory of Magnitudes , Unpublished typewritten documentdrawn from a series of lectures on geometry given at the University of Chicago in 1958-59.41ereni, A. (2019) On the Philosophical Significance of Frege’s Constraint,
Philosophia Math-ematica , 27(2): 244–275.Shapiro, S. & Weir, A. (2000) ‘Neo-Logicist’ Logic is not Epistemically Innocent,
PhilosophiaMathematica , 8: 160–189.Simons, P. (1987) Frege’s theory of real numbers,
History and Philosophy of Logic , 8: 25–44.Shapiro, S. & Snyder E. (2020) Frege on the real numbers, In M. Rossberg and P. Ebert (eds),
Essays on Frege’s Basic Laws of Arithmetic , Oxford, Oxford University Press: 343–383.Wright, C. (2000) Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege’sConstraint,