aa r X i v : . [ m a t h . L O ] J a n THE POTENTIAL IN FREGE’S THEOREM
WILL STAFFORD
Abstract.
Is a logicist bound to the claim that as a matter of analytic truth there is an actualinfinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to beyes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity wasposited. However, this project was abandoned due to apparent failures of cross-world predication. Were-explore this idea and discover that in the setting of the potential infinite one can interpret first-orderPeano arithmetic, but not second-order Peano arithmetic. We conclude that in order for the logicistto weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken themathematics they recover.
Contents
1. Introduction 11.1. Potentially Infinite Models 11.2. Main Results 51.3. A Diversity of Modal Logicisms 61.4. Outline of paper 82. Definitions for a Modal Grundlagen 82.1. Some useful results 103. Proving Modalized Robinson’s Q I PI §
1. Introduction.1.1. Potentially Infinite Models.
In the non-modal setting, Frege (1893; Heck, 1993)essentially proved that second-order Peano arithmetic, PA , is interpretable in the theory HP , which consists of the Second-order Comprehension Schema and Hume’s Principle: ∀ X, Y ( X = Y ⇔ ∃ bijection f : X → Y ) . (HP) I would like to thank the audience at the Logic Colloquium 2016 in Leeds and the UCI Logic Seminar2017 for their questions and comments, and Tim Button, Jeremy Heis, Richard Mendelsohn, Stella Moon,Sean Walsh, and Kai Wehmeier for their helpful feedback.1
WILL STAFFORD
Hume’s Principle characterises the cardinality operator
Theorem . There is a translation from the language of PA to thelanguage of HP that interprets PA in HP . The formal definition of the theories mentioned here can be found in Appendix A. Frege’sTheorem has traditionally been regarded as philosophically important because it is supposedto show that we can derive all arithmetical theorems from an epistemically innocent system.This requires that Hume’s Principle is analytic. However, on the usual semantics, Hume’sPrinciple is only true on domains with at least a countable infinity of objects. This commitslogicists like Frege to the analytic existence of an actual infinity of objects (Boolos, 1998,pp. 199, 213, 233; Hale and Wright, 2001, pp. 20, 292, 309; Cook, 2007, p. 7).A commitment to a potential infinity, in contrast, isn’t a commitment to how many thingsthere actually are, just how many are possible. This is a much safer area in which to makeanalytic claims. Here we show that some but not all of the mathematics of the actualinfinite is recoverable in the setting of the potential infinite. And so, to avoid problematicontological commitments the logicist must also weaken the mathematics they recover.To do this we must decide how to represent Hume’s Principle. Below we will define ‘thenumber of’ operator ✷ ∀ X, Y ( X = Y ⇔ ∃ bijection f : X → Y ) , plus a principle to rigidify the However,we leave the details of this approach for future work. As the modification is so minimal,the move to the potentially infinite doesn’t undermine the justifications offered for Hume’sPrinciple. The syntactic priority thesis can still be argued for as we can identify the be-haviour of terms in a modal setting as well as in a non modal setting. Similarly if we thinkthat abstraction principles offer implicit definitions then this justification works as well inthe modal setting.The rigidity of the octothorpe is important for the success of the project here. However,by assuming that it is rigid we are presuming that ‘the number of’ operator is rigid. Whetherthis is the case in natural language is an empirical question (e.g. Stanley, 1997). We do notaddress this issue here, but two things are worth noting. First the question of the rigidityof ‘the number of’ is not the same question as e.g. whether the number of planets variesbetween worlds. This is because we do not apply the operator to predicates but rather tosets which do not vary their membership across worlds. The second is that this setting doesrule out the possibility of multiple different number structures in the different worlds, e.g.the numbers being von Neumann ordinals in one world and Zermelo ordinals in another.This means that a certain kind of referential indeterminacy which has a prominent place in For those familiar with hybrid systems the axioms needed is ↑ ✷ ∀ X, y ↓ [ X = y → ✷ X = y ] . However, this will not play a role in what follows.
HE POTENTIAL IN FREGE’S THEOREM potentially infinite models . This idea comes from Hodes (1990, p. 379), although he doesnot place exactly these constraints on the accessibility relation. We want the models tobe nearly linear sequences of worlds (if there are two worlds neither of which accesses theother, there is a third world they both access), where later worlds are possible from theperspective of earlier worlds but not the other way around. Each of these worlds shouldcontain only a finite number of objects as we are assuming actual infinities are impossible,and the number of objects should increase from one world to the next. Each world willhave its own second-order domain, which as the worlds are finite, will be the full powerset.The octothorpe will implement Hume’s Principle by taking sets of the same cardinality toa unique object and this object will not change from one world to the next. We define themodels formally as follows:
Definition . A potentially infinite (PI) model is a quadruple M = h W, R, D, I i in themodal signature with second-order quantification and with a as the only non-logicalsymbols, such that the following conditions are met:1.2.1. W is countably infinite and R is a directed partial order, w , written D ( w ), is non-empty and finite for all w ∈ W ,1.2.3. for each n ≥
1, the range of the second-order n -ary relational quantifiers at w is P ( D ( w ) n ) consisting of all subsets of the n -th Cartesian power ( D ( w )) n of D ( w ),1.2.4. if w, s ∈ W such that R ( w, s ) and w = s , then D ( w ) ( D ( s ) , a : ω → D (where D is S w ∈ W D ( w )) assigns to each number n a distinctelement a n in one of the first-order domains, and for all w ∈ W , the cardinality of X is n if and only if X = a n at w . More formally, for w the interpretationfunction is defined as follows: I ( , w ) = {h X, a | X | i | ∃ s ∈ W X ∈ P ( D ( s )) } . Remark . Three brief remarks on this definition:First, conditions 1.2.1-4 define a PI model as a directed partial order of ever-increasingfinite domains. This means that if we have several objects existing in different possibleworlds we can always move to a world where they all exist.Second, condition 1.2.5 defines the cardinality operator | X | . It is sufficient for Hume’s Principle to hold that P ( D ( w ) ) from 1.2.3 is because the quantifier over graphs of functions in Hume’sPrinciple ranges over this set.Third, condition 1.2.5 also ensures that the interpretation of the octothorpe is rigid. Thatis, the octothorpe is interpreted as the same relation at every world. Because of this nothingwill be lost if we write X = x and don’t specify the world of evaluation. In fact, whilewe define X using the a i ’s, we could have instead simply defined it as rigid and satisfyingHume’s Principle and this along with directedness would ensure the a i ’s exist.This definition can obscure the simplicity of the idea here, as such it helps to give severalexamples. The simplest potentially infinite model we can construct is the following: Example . The minimal potentially infinite model is ( ω, ≤ , D, I ) where D ( n ) = { , . . . , n } and the interpretation function I interprets octothorpe as cardinality in the An order R is directed if for all w, s ∈ W there exists an t ∈ W such that R ( w, t ) and R ( s, t ). WILL STAFFORD (a)
The minimal model · · · · · · · · · · · · · · · · · · (b)
The subset model
Figure 1.
Examples of potentially infinite modelsmetalanguage. That is, I ( , w )( X ) = n if and only if | X | = n . The minimal model isillustrated in Figure 1a. When working with such a model we see that a number can bemissing from a world even if a set of that cardinality is present. So I ( , )( { } ) = and ∈ D ( ), but I ( , )( { , } ) = and / ∈ D ( ) even though { , } ⊆ D ( ).A less simple but similarly elementary model makes use of the non-empty finite subsets ofthe natural numbers. This model helps illustrate a non-linear R relation: Example . Let the subset model be ( P ( ω ) <ω − { ∅ } , ⊆ , D, I ) where D ( X ) = X andagain the octothorpe is cardinality. The subset model is illustrated in Figure 1b. Notethat if we have worlds X , . . . , X n we can always find an accessible world whose domainis S ni =0 X i . For example, { , } , { } , { , . . . , } are all finite subsets of the naturalnumbers, none of which access each other, however, their union { , , , , . . . , } isalso a world, which they all access.It is easy to generate unintended models from these two cases. Using the minimal model,for example, we can define the - swap model: Example . The - swap model takes and in the domain of the minimal model andswitches them around. So D ( ) = { } , D ( ) = { , } , D ( ) = { , , } , D ( ) = { , , , } and then for all n ≥ , we have that D ( n ) exactly as it is in the minimal model.These models should help illustrate the intuition behind the potentially infinite models.They will also be helpful when we need counterexamples to claims later in the paper.We can now define satisfaction for potentially infinite models using a standard semanticsfor quantified modal logic, such as in Fitting and Mendelsohn (1998). Three things to notefirst: (1) Our quantifiers are actualist, but free variables may be assigned to objects in anyworld. (2) Set variables are interpreted rigidly across worlds. That is the membership of aset doesn’t change depending on the world. (3) To simplify the notation, instead of variableassignments, we work as though we had a rigid name for every object in the models. Recallthat M , w (cid:15) ϕ means that given any replacement of free variables with the added constantswe evaluate ϕ as true in M at world w . With this in place, the notion of potentially infinitemodels induces a natural validity relation, which we define as follows: Definition . We say that ϕ is true in all potentially infinite models , or (cid:15) PI ϕ , if forall potentially infinite models M and worlds w ∈ W we have M , w (cid:15) ϕ . We define ϕ (cid:15) PI ψ as for all models M and worlds w ∈ W , if M , w (cid:15) ϕ then M , w (cid:15) ψ .The consequence relation here is defined locally rather than globally (Fitting and Mendel-sohn, 1998, p. 21). This is because the deduction theorem holds for the local consequencerelation but not the global one (Fitting and Mendelsohn, 1998, p. 23). I will use bold face numbers for the numbers in the metalanguage.
HE POTENTIAL IN FREGE’S THEOREM We will now state our two main results which together show that wecan interpret the first-order theories of first-order Peano arithmetic PA and first-order truearithmetic TA , but not the second-order theories of second-order Peano arithmetic PA andsecond-order true arithmetic TA , in theories defined in terms of potentially infinite models.A deductive theory for second-order modal logic with rigid operators would be unwieldyand the complications caused by it would be likely to obscure the insights provided bythe Kripke semantics. Hence, we leave development of a deductive theory for future work.We can define a theory just in terms of the potentially infinite models. This theory will bestronger than anything we could produce deductively because it does not admit nonstandardmodels of the natural numbers. Because of this we will call it the external theory of thepotentially infinite or E PI : E PI = { ϕ | (cid:15) PI ϕ } . (1)To capture something closer to what can be deduced from the models we need to usethe model-theoretic validity relation defined above, relativised to a weak metatheory. Thetheory ACA is a subsystem of PA which only has comprehension for first-order formulas.More information about this theory can be found in Appendix A. Since we can code finitesets of natural numbers as natural numbers in ACA , we can define the property of being apotentially infinite model in this theory, along with the associated validity notion (cid:15) PI . Thisgives us the internal theory of the potentially infinite or I PI : I PI = { ϕ | ACA ⊢ ‘ (cid:15) PI ϕ ’ } . (2)Intuitively, this theory is every formula that can be proven valid on potentially infinitemodels, given the weakest metatheory that can formalise the models. A full definition isgiven in Appendix B. The definition of interpretation is traditionally restricted to theoriesin the same logic, whereas in this setting E PI and I PI are theories in second-order modal logicbut PA , PA , TA , and TA aren’t modal theories. So, to state and prove our main resultswe need a more general notion of generalised translation and interpretation which capturesthose interpretations which involve not just different theories but different logics. This isdefined in section 5. Our first main result is: Theorem . (i) There is a generalised translation from the language of PA to thesecond-order modal language with octothorpe that interprets TA in E PI .(ii) There is a generalised translation from the language of PA to the second-order modallanguage with octothorpe that interprets PA in I PI . Further, this is a PA -verifiablegeneralised interpretation. This result is proven in Section 5. The translation used is based on one offered by Linnebo(2013) in the setting of modal set theory. The key difference, compared with the standardnotion of translation, is that “for all” is translated as “necessarily for all” and, similarly,“there is” is translated as “possibly there is.”The first theorem shows that the PI models capture a significant amount of mathematics.However, we cannot strengthen the result to second-order theories of arithmetic as oursecond main theorem shows:
Theorem . (i) There is no generalised translation from the language of PA to thesecond-order modal language with octothorpe that interprets TA in E PI . We picked the weakest theory because we are interested in what is deducible from PI models and if westrengthen the metatheory I PI will be strengthened in ways that reflect what the metatheory thinks aboutfinite sets (which can code consistency statements). WILL STAFFORD (ii) There is no generalised translation from the language of PA to the second-order modallanguage with octothorpe that PA -verifiably interprets PA in I PI . For both E PI and I PI , the results follow from the fact that PI models are Π definable. Andthis follows because all of the worlds are finite. Because of this, PI models are representablein reasonably weak theories of second-order arithmetic. But then limitive results aboutwhat theories can represent about themselves will stop theories that can represent E PI and I PI being interpretable into E PI and I PI .These results are important because they show that less mathematics is analytic on thephilosophical perspective which motivates the potentially infinite models than on the tradi-tional perspective. The external theory cannot recover TA but only TA . And the internaltheory cannot recover PA but only PA . Further, PA has traditionally been the target ofFregean interpretation results as it allows for the recovery of analysis and much of mathe-matics. Analysis can be coded in second-order Peano arithmetic, as real numbers can becoded as sets of rationals, which in turn can be coded as naturals. This means that Frege’stheorem already accounts for a larger expanse of mathematics than it might first appear. Ifwe try to avoid the claim that it is analytic that there are actually infinitely many objects,however, it then seems we will not have managed to recover as much mathematics. If weare looking to show that mathematics is analytic, we have moved further from our goal.However, we have still captured a substantial chunk of our most frequently used math-ematics. Feferman (2005, p. 613) has argued that all scientifically applicable analysis canbe developed in PA or a conservative extension of it. If this is correct then we can stillrecover the mathematics for which an explication of its truth is most philosophically fruitful,namely the mathematics which we rely on when we act in the world. One might wonder whya logicist would care about whether or not the mathematics recovered is used. But it seemswe should keep an open mind to different parts of mathematics being justified in differentways. Maybe something as fundamental as first-order arithmetic turns out to be analytic,but it seems unlikely that the same is true of the higher reaches of set theory. With this inmind, it should not be damaging that not all mathematics turns out to be analytic.
The idea of using the potentially infinite as afoundation of logicism has a pedigree in the work of Putnam and Hodes, and more recentwork on modal foundations of mathematics and on variants of Frege’s theorem in differentlogics. Putnam suggested that by accepting a modal picture of mathematics we could avoidbeing Platonists about the numbers or committing to how many objects there actually are.This is stated most clearly when he writes:‘Numbers exist’; but all this comes to, for mathematics anyway, is that (I) ω -sequences are possible (mathematically speaking); and (2) there are necessarytruths of the form ‘if α is an ω -sequence, then . . . ’[.] (Putnam, 1967, pp. 11–12)Hodes took on this idea, but he was sceptical of the existence of actual infinities. He thoughtthat ‘[a]rithmetic should be able to face boldly the dreadful chance that in the actual worldthere are only finitely many objects’ (Hodes, 1984, p. 148). His solution made use of theidea of the potentially infinite rather than the actually infinite. He appealed to modality Demopoulos (1994, 238 n26) points out that Frege often uses arithmetic when he means somethingbroader including analysis. For example, “By the fact of the proof-theoretical reduction of W to [ PA ], the only ontology it commitsone to is that which justifies acceptance of [ PA ].” (Feferman, 2005, p. 613) Feferman works in a system W which contains types for the naturals, the cross product and partial functions. The full classical analysis ofcontinuous functions can be carried out in W . (Feferman, 2005, p. 611) HE POTENTIAL IN FREGE’S THEOREM v , ✸ ( ∃ v )( N ( v )& . . . ) “moves us” to other worlds u and then has us seeka witnessing member of [the natural number in the model] in [the domain of u ];we may find one, but then have no way “back” to w to see what hold [sic] for itthere. (Hodes, 1990, p. 388)So we might know that there possibly exists a number with a property, but in Hodes’ssystem, we have no way of returning to our original world to use what we have found. Forexample, if we find the number of a set in some world, we have no assurance that thisnumber is available for us to talk about in the world the set came from. It is only knownthat it is the number of the set in the world the number exists in . The difficulty identifiedhere is with cross-world predication, which occurs when we want to say something about anobject in one world and how it relates to objects in another world (Kocurek, 2016).In what follows we will show that the problem is not with cross-world predication per se .Both by working directly with the models, but also by allowing the octothorpe to be rigid,we can mimic some of the effects of cross-world predication. Yet in this setting we recoversome but not all of the arithmetic recovered by Frege’s theorem. Indeed, our main results,Theorems 1.8 and 1.9, show that the situation is more complicated than Hodes suggested,and that a partial realisation of his project is possible.There are two recent trends in the study of logicism which this project is connected to.First, Studd (2016) has suggested that the modal setting is an attractive one for the logi-cist because it would help to solve the bad company objections. Unlike here, Studd’s isconcerned with inconsistent abstraction principles and in particular set abstraction. Thisis interestingly connected to the na¨ıve conception of set because one can think of the un-restricted set Comprehension Schema as similar in spirit to a modal version of Basic LawV. While work in this area goes back to Parsons (1983), it has been pursued recently byLinnebo (2013; 2018). Much of Linnebo’s work has been on set theory. The concerns thereare very different from ours, as it make little sense in set theory to worry about the actualinfinite not existing and set theory is generally treated in first-order logic.The work in thispaper takes inspiration from the results presented in Linnebo (2013) and (2018) and makesuse of a similar method of translating between the modal and non-modal setting. How-ever, while the dynamic abstraction principles discussed by Linnebo (2018) resemble thebehaviour of the number of operator, his preferred abstraction principle for arithmetic isordinal abstraction (Linnebo, 2018, Ch. 10.5), whereas in this paper we work with a modalversion of Hume’s Principle, a cardinality principle.Second, there has been a lot of recent work on whether Frege’s Theorem still holds whenthe logic is modified in certain ways. Bell (1999) and Shapiro and Linnebo (2015) haveshown that Frege’s Theorem is available in the intuitionistic setting. Burgess (2005) andWalsh (2016) found that a version of Frege’s Theorem is possible in a certain predicativesetting. Kim (2015) proves a version of Frege’s Theorem in a modal setting. This employsan axiomatised version of the ‘the number of F ’s is n ’ as a binary relation, instead of thetraditional type-lowering ‘number of’ operator. Kim recovers the axioms of PA but findsthat a restricted version of HP holds. The modality used is S5 and meant to representlogical possibility, not potentiality. Because of this Kim’s system does not have the same WILL STAFFORD structure of our models, where the numbers slowly grow. Closer in spirit to the workhere is that on finite models of arithmetic by Mostowski (2001). There he considers initialsequences of the natural numbers and what holds over all such models. These have a clearconnection to the minimal model discussed above. Urbaniak (2016) has taken Mostowski’smodels and worked with them in a modal setting. They have shown that Le´sniewski’s typed,free logic with modal quantifiers, which proves a predicative version of HP , can interpret PA . Our setting is quite different from that of Urbaniak’s paper as Le´sniewski’s typed, freelogic differs dramatically from the one we work in here. The work in this paper proceedsby looking at whether a version of Frege’s Theorem is available in a classical second-ordermodal setting. Unlike these other results, we find that a modal version of Frege’s Theoremfor PA is not possible, as shown by Theorem 1.9. This paper is organised as follows. Section 2 expands thepotentially infinite models’ language to include the language of arithmetic. In Section 3we show that using the expanded language the potentially infinite models satisfy a weaktheory of arithmetic equivalent to a modal version of Robinson’s Q . In Section 4 we define theinductive formulas of the language and show that induction holds for them. This allows us toshow Theorem 1.8, that TA is interpretable in our external theory and PA is interpretablein our internal theory, in Section 5. In Section 6 we show that no natural interpretation of PA is possible by proving Theorem 1.9. §
2. Definitions for a Modal Grundlagen.
Just as Frege in the
Grundlagen definedthe numbers and the relations on them using only the ‘number of’ operator, here we showhow modified versions of Frege’s definitions can do this in the setting of the potentiallyinfinite. Proving that these definitions satisfy the usual arithmetical axioms will occupyus in §§ Grundlagen ,the definitions will be intuitive and correspond to our understanding of cardinal numbers.The first definition is easy and does not require any of the modal apparatus. We simplylet 0 = ∅ . This follows Frege (1884, §
74 p. 87) explicitly, who said that zero is “theNumber which belongs to the concept ‘not identical with itself’”.Next we must define the successor, as the other definitions rely on it. The definition hereis like the one offered by Frege, but it differs by allowing the sets which witness that oneobject is the successor of another to be merely possible. This is to ensure that if an objectis ever the successor of another, then it is the successor of that object in every world wherethey both exist. This property will be important in the proof of induction. The definition ofsuccessor, in plain terms, is: one object is the successor of another just in case it is possiblethat there are two sets, which differ by one object and the successor is the number of thelarger set, and the predecessor is the number of the smaller set. Figures 2a and 2b illustratethe two ways this can be done, resulting in two definitions of the successor:
Definition . Sxy ≡ ✸ ∃ G, u [ Gu ∧ ( y = G ) ∧ ( x = G − { u } ))](3) This has some precedent in Hodes (1990, p. 383). However, whereas we (and Frege) first define successorand then use this to build the other definitions, Hodes takes ‘less than or equal to’ as his primitive. In hissystem a number N (understood as a higher-order object) is less than or equal to another number N ′ justin case it is possible that there are two other second-order objects A and A ′ each with the same numberof objects as N and N ′ respectively and A is a subset of A ′ . That this has parallels with the definition ofsuccessor offered here will be clear on inspection. HE POTENTIAL IN FREGE’S THEOREM ∃ ab Xx Sab a = X/ { x } b = X (a) Diagram of when a is suc-ceeded by b. ∃ ab X x Sab a = Xb = X ∪ { x } (b) Diagram of when a is suc-ceeded by b for alternativedefinition of S. ∃ ab c X Y+( a, b, c ) a = Xb = Yc = X ∪ Y (c) Diagram of when c is theaddition of b and c.
Figure 2 S ′ xy ≡ ✸ ∃ F, u [ ¬ F u ∧ ( x = F ) ∧ ( y = F ∪ { u } ))](4)The first of these definitions simply adds the possibility operator to the definition of successorsuggested by Frege (1884, §
76 p. 89). These definitions are equivalent: to see this, simplyconsider F = G − { u } and G = F ∪ { u } . In what follows we will simply use the definitionthat is most convenient and will write S for both.The definition of addition is similarly intuitive. The relation + holds between three objects a , b , and c such that it is possible that there are disjoint sets X and Y of cardinality a and b respectively, and c is the cardinality of X ∪ Y , the union of the two disjoint sets. This isillustrated by Figure 2c and can be written formally as: Definition . +( a, b, c ) ≡ ✸ ∃ X, Y ( a = X ∧ b = Y ∧ c = X ∪ Y ∧ ( X ∩ Y ) = ∅ )(5)For c to be the result of multiplying a and b we need a set B of cardinality b and for eachelement x of B a set A x of cardinality a . The A x ’s must all be disjoint. And c must be thecardinality of the union of all the A x ’s. To define the A x ’s we define a binary relation P that holds between x in B and all y in A x . So A x is { y | P xy } . Definition . (6) × ( a, b, c ) ≡ ✸ ∃ X, P [ X = b ∧ ∀ x ∈ X ( { y | P xy } = a ) ∧ ∀ x, y ∈ X ( x = y → { z | P xz } ∩ { z | P yz } = ∅ ) ∧ [ x ∈ X { y | P xy } = c ]The definition of the natural numbers is more complicated and require us to define thenotion that one number follows another in the ordering of the natural numbers. We willmake use of Frege’s definition from the 1879 Begriffsschrift (1967, § III pp. 55 ff; 1884, §
79 p. 92 ff). Russell and Whitehead (1910, p. 316) called this relation the ancestral relation because a good example of what it does is define the relation ‘ancestor of’ from the relation‘parent of’. The strong ancestral of ϕ holds between two objects a and b just in case b iscontained in every set such that the set is closed under ϕ and the set contains everything a bears ϕ to. So, we can define someone’s ancestors as everyone who is in every set thatcontains their parents and the parents of everyone in the set. It is not guaranteed that a bears this relation to itself, and so we also define the reflexive weak ancestral . For easy of readability, we will use set theoretic notation as a convenient short hand for concepts formedusing the language of the model. So F ∪ { u } is used for the concept given by Xx ↔ ( F x ∨ x = u ). WILL STAFFORD
Definition . ϕ + ( a, b ) ≡ ∀ X [( ∀ x, y ( Xx ∧ ϕ ( x, y ) → Xy ) ∧ ∀ x ( ϕ ( a, x ) → Xx )) → Xb ] . Definition . ϕ += ( a, b ) ≡ ϕ + ( a, b ) ∨ a = b. Using this definition, we define a natural number as an object that is some finite number ofsuccessor steps from 0, assuming 0 exists.
Definition . N x ≡ S += x ∧ ∃ y ( y = 0) . This definition closely parallels Frege’s, though the definition of S is different. The existenceclaim is added because in the modal setting 0’s existence cannot be assumed. For example,0 does not exist at worlds , , and in the - swap model, and, as is not a memberof infinitely many finite subsets of the natural numbers, 0 does not exist at infinitely manyworlds in the subset model. In these worlds nothing is a natural number. The following six lemmas will help explain the behaviour of N in the models. We admit the proofs as they do not pose any particular difficulty. For thefollowing Lemmas, recall Definition 1.7 where (cid:15) PI ϕ was defined as ϕ is true in all worlds inall potentially infinite models. First, note that the set defined by N at a world satisfies theantecedent of S + x . Intuitively, the idea here is that if x is in every set containing 0 andclosed under S , and Sxy , or S y , then y must also be in every set with these properties. Lemma . (cid:15) PI ∃ x ( x = 0) → ∀ y ( S y → N y )) Lemma . (cid:15) PI ∀ x, y ( N x ∧ Sxy → N y )It follows immediately from this that if x exists at a world and at that world N y and Syx then N x . However, that doesn’t mean N is the set of all numbers across all worlds as N only holds of objects which exist at the world of evaluation. This contrasts with our otherdefinitions where the objects need not exist at the world. Lemma . | = P I N x → ∃ y y = x This is because the quantifiers in N are plain rather than having modals in front of them.This is important because if we put the modals in front everything is a number!We informally extend our definition of the interpretation function I to I ( N , s ) = { x ∈ D ( s ) | M , s (cid:15) N x } . Note that by Lemma 2.9 we have { x ∈ D ( s ) | M , s (cid:15) N x } = { x ∈ D |M , s (cid:15) N x } , where D is the domain of the model not the world.Recall that a i is the unique element in D such that if | X | = i then I ( , w )( X ) = a i asdefined in 1.2.5. We can now explicitly describe the interpretation of N at a world w interms of the a i ’s, that is, the set I ( N , w ): Lemma . Let w be a world and let n be the first number such that a n / ∈ D ( w ) . Thenif n > , it follows that { , a , . . . , a n − } = I ( N , w ) , and further, n = 0 iff I ( N , w ) = ∅ . This result shows us how the differences between our modal setting and the traditional non-modal setting of the
Grundlagen become most stark in the case of the interpretation of thenatural numbers at a world. Two things are worth highlighting. The first is that N is finiteat every world, since it is a subset of the domain of the world, and the domain of every worldis finite. The second is that objects that are not in N at one world can ‘become’ numbersat later worlds. This doesn’t happen in the minimal model, where I ( N , n ) = D ( n ) at every HE POTENTIAL IN FREGE’S THEOREM I ( N , { , } ) = ∅ , I ( N , { , , } ) = { , } and I ( N , { , , , , } ) = { , , , } . This distinguishes ¬ N ( x ) from the otherrelations which have a certain stability; if objects stand in these relations at one world,then they do so in all worlds in which they all exist. The formal definition of stability isgiven as Definition 8. This difference is caused by there being no possibility operator at thebeginning of the definition of N . Despite this, once something is a number it remains one: Lemma . (cid:15) PI S ( x, y ) → ✷ S ( x, y ) holds, as does (cid:15) PI S + ( x, y ) → ✷ S + ( x, y ) , (cid:15) PI S += ( x, y ) → ✷ S += ( x, y ) and (cid:15) PI N x → ✷ N x . It is also worth noting that even though some cardinalities may not be numbers at everworld, the cardinality of every set eventually becomes a natural number.
Lemma . For all w ∈ W and X ⊆ D ( w ) , there is a world s such that R ( w, s ) and X ∈ I ( N , s ) . This is because §
3. Proving Modalized Robinson’s Q . In what follows we will prove that the modal-ized axioms of Robinson’s Q are true on all PI models (cf. Definition 1.7). Robinson’s Q is a weak theory of arithmetic that defines successor as an injective function that neverreturns 0 and gives a recursive definition of addition and multiplication. By “modalized”we mean that we write “necessarily for all” for “for all” and “possibly there is” for “thereis”. In other words, it is what results when we apply the Linnebo translation (mentionedin the introduction) to the axioms of Robinson’s Q . The theory PA is obtained by addingthe mathematical induction schema to Q . We deal with PA and the proof of the inductionschema in Section 4. First we will show that our relations define the graphs of functions. The easiest case issuccessor.
Lemma . (cid:15) PI ✷ ∀ x, y, z ∈ N (( Sxy ∧ Sxz ) → y = z ) . Proof.
Let s ∈ W and x, y, z ∈ I ( N , s ) satisfy the antecedent. As x is the predecessor inboth relations it follows by directedness that there is a w ∈ W , such that R ( s, w ) where thereare X, X ′ ⊆ D ( w ) and X = x = X ′ . As such there is a bijection g : X → X ′ . Therewill also be a, b ∈ D ( w ) such that a / ∈ X , b / ∈ X ′ , and y = X ∪ { a } and z = X ′ ∪ { b } .As a / ∈ X and b / ∈ X ′ we can construct h such that for all u ∈ X , h ( u ) = g ( u ) and h ( a ) = b .Clearly h is a bijection, so y = X ∪ { a } = X ′ ∪ { b } = z . ⊣ Lemma . (cid:15) PI ✷ ∀ x ∈ N ✸ ∃ y ∈ N Sxy . Proof.
As illustrated in Figure 3a, let s ∈ W and x ∈ I ( N , s ), it follows that x = a n for some n and, by Lemma 2.10, { , . . . a n − } ( D ( s ). Further, a n = { , . . . a n − } and a n / ∈ { , . . . a n − } . Thus, there must be a further world w accessible from w and a y ∈ D ( w )such that y = { , . . . a n − } ∪ { a n } . It follows that Sxy at w . By Lemma 2.11 x ∈ I ( N , w ).As N is closed under successor by Lemma 2.8, we have that y ∈ I ( N , w ). And since R istransitive, w is accessible from s . ⊣ A list of the non-modalized axioms can be found in Appendix A. While what we show here is that theseaxioms are in the theory E PI , each of the proofs that follow can be formalised in ACA (cf. Appendix B).That this is possible will ensures that all axioms proven here are also in the theory I PI (from Section 1.2).This is a key point in the proof of Theorem 1.8.ii which we complete in section 5. WILL STAFFORD ∃ ∃ ∃ x yaX X x = X y = X ∪ { a } (a) Finding the successor of x Y g o f f x y X N X N (b) Proof of the recursion clause for addition
Figure 3
These two proofs offer a general outline of the reasoning for addition and multiplication.For S1 this strategy is to show that whatever x is the sets assigned to y and z will havethe same cardinality. Where as for S2 one simply needs to construct a set of the correctcardinality. For this reason we do not give the proofs for the next four lemmas. Lemma . (cid:15) PI ✷ ∀ x, y, z, z ′ ∈ N (+( x, y, z ) ∧ +( x, y, z ′ ) → z = z ′ ) . Lemma . (cid:15) PI ✷ ∀ x, y ∈ N ✸ ∃ z ∈ N + ( x, y, z ) . Lemma . (cid:15) PI ✷ ∀ x, y, z, z ′ ∈ N ( × ( x, y, z ) ∧ × ( x, y, z ′ ) → z = z ′ ) . Lemma . (cid:15) PI ✷ ∀ x, y ∈ N ✸ ∃ z ∈ N × ( x, y, z ) . We also need to show that 0 meets the right conditions to be a constant.
Lemma . (cid:15) PI ✸ ∃ x ∈ N ( x = 0 ∧ ✷ ∀ y ( y = 0 → y = x )) . Proof.
By the definition of N , it follows that 0 ∈ I ( N , s ) for any world s with 0 in thedomain. And as 0 = ∅ there is some s with 0 in the domain. The second conjunct followsby the transitivity of identity. ⊣ We can now move on to the recursion equations in Q . We separate these into the base stepsconcerning 0 and the recursive step. For the base steps, because 0 = ∅ the proofs of thelemmas are relatively straight forward. As such we list them here without proof. Lemma . (cid:15) PI ¬ ✸ ∃ x ∈ N ( Sx . Lemma . (cid:15) PI ✷ ∀ x ∈ N + ( x, , x ) . Lemma . (cid:15) PI ✷ ∀ x ∈ N × ( x, , . What is left now is to show the recursion steps. He we only prove the case for + as one canuse the same stratagy for × and the proof is simple for S . Lemma . (cid:15) PI ✷ ∀ x, y, z ∈ N (( Sxz ∧ Syz ) → x = y ) . The proof simply follows from the fact that if there is a bijection between two sets X and Y then there will be a bijection between X ∪ { a } and Y ∪ { b } if a and b aren’t in X or Y respectively. Lemma . (cid:15) PI ✷ ∀ n, x , x , y , y , z ∈ N ( S ( x , x ) ∧ S ( y , y ) ∧ +( n, x , y ) ∧ +( n, x , z ) → y = z ) . Proof.
As illustrated in Figure 3b, let s ∈ W and n, x , x , y , y , z ∈ I ( N , s ) satisfy theantecedent. We want to show that y = z . By directedness, we know there is a world w containing all the objects and sets which the antecedent states possibly exist. As y succeeds y there is a set Y and an object a / ∈ Y at w such that y = Y ∪ { a } and y = Y . HE POTENTIAL IN FREGE’S THEOREM y to be the addition of n and x so there are disjoint sets N and X such that n = N , x = X , and y = N ∪ X . Further there is a bijection g : Y → N ∪ X .Now let b be an element not in N or X (we can always pick w so that such an elementexists). Clearly we can define a bijection o between the singletons of a and b . Now, using g and o , define the bijection g : Y ∪ { a } → N ∪ X ∪ { b } , as the union of g and o . Now as x is the successor of x , it follows that x = X ∪ { b } . As z is the addition of n and x there are disjoint sets N ′ and X such that n = N = N ′ , x = X ∪ { b } = X and z = N ∪ X . As such there are bijections f : X ∪ { b } → X and f : N → N ′ . So, wecan define the bijection f : N ∪ X ∪ { b } → N ′ ∪ X as f on X ∪ { b } and f on N . Thenas z = N ∪ X the composition f ◦ g is a bijection proving y = z . ⊣ Lemma . (cid:15) PI ✷ ∀ n, x , x , y , y , z ∈ N ( S ( x , x ) ∧ +( n, y , y ) ∧ × ( n, x , y ) ∧× ( n, x , z ) → y = z ) . This proof is similar to the above except we end up showing that y = S x ∈ A ∪{ u } { y | P xy ∨ ( x = u ∧ y ∈ N ) } = S x ∈ A { y | T xy } = z where A , A , and N are of cardinality x , x , and n respectively and P is the relation given by × ( n, x , y ) and T by × ( n, x , z ).These results show that we have successfully defined a modalized version of Robinson’s Q in our system. The next section will recover a modalized induction schema. §
4. Proving the Modalized Induction Schema.
We have succeeded in giving a weaktheory of arithmetic in a potentially infinite setting. However, we can recover more arith-metic by proving that when restricted to appropriate formulas a modalized version of theinduction schema is true on all PI models. The modalized induction schema is:[ ϕ (0) ∧ ✷ ∀ x, y ∈ N ( ϕ ( x ) ∧ S ( x, y ) → ϕ ( y ))] → ✷ ∀ x ∈ N ϕ ( x )(7)Modalized induction does not hold for all formulas in our models, as will be shown inLemma 4.4. So, we need to define a subclass of the formulas in the language of potentiallyinfinite models for which it does hold. These we will call the inductive formulas, and inLemma 4.3 it will be proven that induction does hold for inductive formulas. Definition . The inductive terms and formulas are defined recursively as follows:1. An inductive term is either 0 or a first-order variable.2. If t , t , t are inductive terms then t = t , S ( t , t ), +( t , t , t ) and × ( t , t , t ) areinductive formulas.3. Applications of the propositional connectives to inductive formulas are inductive for-mulas.4. If ϕ is an inductive formula then ✷ ∀ x ∈ N ϕ and ✸ ∃ x ∈ N ϕ are inductive formulas.The inductive terms and formulas are a subset of the terms and formulas respectively. Anyterm of the form X is not an inductive term, and indeed no term or formula with a freesecond-order variable is inductive. Likewise N ∀ z ( x = z ) and ∃ y ( S y ) are not inductiveformulas, while ✷ ∀ z ∈ N ( x = z ) and ✸ ∃ y ∈ N ( S y ) are.A formula ϕ is stable when: (cid:15) PI ϕ → ✷ ϕ. (8) This terminology is used to distinguish between these formulas and other for which induction does nothold. Hopefully no confusion will be caused by the distinct uses of the term inductive formulas elsewherein the literature. WILL STAFFORD
Stability is taken from Linnebo’s (2013, p. 211) work on set theory in a modal setting. Itmeans once a formula has been made true it stays true. As we saw in Lemma 2.11, S , S + ,S += , and N are all stable and an example of an unstable formula is ¬ N . Fortunately, theinductive formulas all have the property of being stable, as we will now prove. This willallow us to prove induction for these formulas. Lemma . If ϕ is an inductive formula then (cid:15) PI ϕ → ✷ ϕ . Proof.
In what follows we prove by induction on the complexity of the inductive formulasthat both ϕ → ✷ ϕ and ✸ ϕ → ϕ . The second condition is included to deal with the case ofnegation.Base case: x = y and x = 0: The result follows from the evaluation of ∅ being rigid andthe identity relation being interpreted as the identity from the metalanguage. Note thatfor S, + , and × that ✸ ψ → ψ follows simply because R is transitive and they start witha ✸ . S ( x, y ): See Lemma 2.11. +( x, y, z ): Assume that M , w (cid:15) +( a, b, c ). It follows thatthere exists a world w ′ accessible from w and nonintersecting sets A, B ⊆ D ( w ′ ) satisfying+. Let s be a world such that R ( w, s ). Then by directedness, there is a world s ′ such that R ( s, s ′ ) and R ( w ′ , s ′ ), and A, B ⊆ D ( s ′ ). So +( a, b, c ) holds at s . × ( x, y, z ): The reasoningis essentially the same as that used for +.Now we proceed to the induction step. We will only show the case of the quantifier as ¬ and ∧ proceed as one would expect. ✸ ∃ x ∈ N ψ : Assume M , s (cid:15) ✸✸ ∃ x ∈ N ψ . It followsby transitivity that M , s (cid:15) ✸ ∃ x ∈ N ψ . Now we show that ( ✸ ∃ x ∈ N ψ ) → ( ✷✸ ∃ x ∈ N ψ ).First take a world w such that ✸ ∃ x ∈ N ψ holds at w . Then take worlds s, w ′ such that R ( w, s ), R ( w, w ′ ), ∃ x ∈ N ψ holds at w ′ and we want to show ✸ ∃ x ∈ N ψ holds at s . At w ′ there is an a ∈ D ( w ′ ) such that a ∈ I ( N , w ′ ) and ψ ( a ) holds at w ′ . So, by Lemma 2.11, N a → ✷ N a holds at w ′ and by the induction hypothesis, ψ ( a ) → ✷ ψ ( a ). Let s ′ be suchthat R ( s, s ′ ) and R ( w ′ , s ′ ), such a world exists by directedness. It follows that N a and ψ ( a )hold at s ′ and as s ′ is accessible from s we have proven ✸ ∃ x ∈ N ψ holds at s . ⊣ We can now prove that the modalized induction schema holds for all inductive formulas.We do this by showing the more general result that induction holds for all stable formulas.
Lemma . If ϕ is stable, then (cid:15) PI [ ϕ (0) ∧ ✷ ∀ x, y ∈ N ( ϕ ( x ) ∧ S ( x, y ) → ϕ ( y ))] → ✷ ∀ x ∈ N ϕ ( x ) . Proof.
Let w be a world. Further, we assume the antecedent of the induction schemaholds so let ϕ (0) and ✷ ∀ x, y ∈ N ( ϕ ( x ) ∧ S ( x, y ) → ϕ ( y )) hold at w . Let s be a worldaccessible from w and let a ∈ I ( N , s ). We will show that ϕ ( a ) at s . If a = 0 then, as ϕ isstable, we are done so assume not.As a ∈ I ( N , s ), if we prove ∀ x, y ( ϕ ( x ) ∧ N x ∧ S ( x, y ) → ϕ ( y ) ∧ N y ) and ∀ x ( S (0 , x ) → ϕ ( x ) ∧ N x ) hold at s then we have satisfied the antecedent of S + a and so it follows that ϕ ( a ) ∧ N a at s .At s we have ∀ x, y ∈ N ( ϕ ( x ) ∧ S ( x, y ) → ϕ ( y )). We also have that if x ∈ I ( N , s ) , and S ( x, y ) hold at s then by Lemma 2.8 that y ∈ I ( N , s ). This proves ∀ x, y ( ϕ ( x ) ∧ N x ∧ S ( x, y ) → ϕ ( y ) ∧ N y ) at s .From a ∈ I ( N , s ) it follows that 0 ∈ D ( s ). Assume x ∈ D ( s ) and S x , as 0 ∈ D ( s ) itfollows by Lemma 2.7 that x ∈ I ( N , s ). It then follows by the stability of ϕ that ϕ (0) at s .As such we have the antecedent of ∀ x, y ∈ N ( ϕ ( x ) ∧ S ( x, y ) → ϕ ( y )) so we get ϕ ( x ). Andfrom this it follows that ∀ x ( S (0 , x ) → ϕ ( x ) ∧ N x ) holds at s . ⊣ HE POTENTIAL IN FREGE’S THEOREM
Lemma . If ϕ ( x ) is ∀ z ( z = x ) , then PI [ ϕ (0) ∧ ✷ ∀ x, y ∈ N ( ϕ ( x ) ∧ S ( x, y ) → ϕ ( y ))] → ✷ ∀ x ∈ N ϕ ( x ) . Proof.
It is sufficient to show there is a model and a world in the model where thisstatement is false. Take the minimal model from Example 1.4 and world , where D ( ) = { } . Clearly M , (cid:15) ∀ z ( z = 0). Let w ∈ W be such that R ( , w ) and assume that forall x, y ∈ I ( N , w ), that ∀ z ( z = x ) and S ( x, y ) hold at w . As everything in the domain isequal to x it follows that y = x and so ∀ z ( z = y ) at w . So M , (cid:15) ✷ ∀ x, y ∈ N ( ∀ z ( z = x ) ∧ S ( x, y ) → ∀ z ( z = y )). But it does not follow that ✷ ∀ x ∈ N ∀ z ( z = x ), because 1 ∈ W is a counterexample as D ( ) = { , } . ⊣ §
5. Proof of Theorem 1.8.
We now have almost all the pieces needed to prove Theo-rem 1.8. However, before we do that we need to discuss what a translation and interpretationare in our setting because we are moving between logics.Intuitively, a translation between two languages starts with instructions on how to rewriteatomic formulas in one language into the other language. It does not make any changesto the propositional connectives but can restrict the quantifiers to objects meeting someconditions. In the current setting, however, we need a formal definition of what is tocount as a translation when the underlying logics are different. This notion should, at thevery least, capture the Linnebo translation. We offer the following definition as a minimalcondition on any translation, though more will need to be done to ensure a widely applicabledefinition of translation and interpretation between logics.
Definition . Let L A and L B be two logics extending first-order predicate logic, de-fined by the languages L A and L B and derivability relations ⊢ L A and ⊢ L B respectively. A generalised translation is given by a recursive map ( · ) G : L A → L B which preserves freevariables and a domain formula δ ( x ) ∈ L B , such that the map is compositional on thepropositional connectives and where for all unnested formulas ϕ , . . . , ϕ n , ψ containingfree variables x , . . . , x m one has the following: ϕ , . . . , ϕ n ⊢ L A ψ ⇒ δ ( x ) , . . . , δ ( x m ) , ϕ G , . . . , ϕ G n ⊢ L B ψ G (9)What we have done so far is an informal translation from the first-order language ofarithmetic into the signature of the potentially infinite models. In Section 2 we showed howthe atomic formulas could be translated. Further, the modalized versions of the axiomsof PA proven in Sections 3 and 4 are the translations of PA ’s axioms via the translationfound in Section 2 and the Linnebo translation for the quantifiers.While it has been set out in previous sections, for the sake of definiteness we here recordthe translation explicitly. We will call this translation ( · ) F , as it is a Fregean translation.Three things are worth noting before we lay out the translation. The first is that the domainformula associated to this interpretation is N from Definition 2.6. The second is that therange of this translation is the inductive formulas from Definition 4.1. The third is that An unnested formula is one where the atomic subformulas of a formula contain at most one constant,function or relation (Hodges, 1993, p. 58). We only give conditions for unnested formulas. So, for example,
Sxy and +( x, y, z ) are unnested but S x and +(0 , , z ) are nested. Every formula is equivalent to an unnestedone (Hodges, 1993, p. 59, Cor 2.6.2). As such the translation can be expanded to unnested formulas usingthis equivalence. WILL STAFFORD · ) F .0 F ≡ ∅ , (10) Sab F ≡ ✸ ∃ G ∃ u [ Gu ∧ ( b = G ) ∧ ( a = G ∪ { u } )] , (11) +( a, b, c ) F ≡ ✸ ∃ X, Y ( a = X ∧ b = Y ∧ c = X ∪ Y ∧ X ∩ Y = ∅ ) , (12) × ( a, b, c ) F ≡ ✸ ∃ X, P [ X = b ∧ ∀ x ∈ X ( { y | P xy } = a ) ∧∀ x, y ∈ X ( x = y → { z | P xz } ∩ { z | P yz } = ∅ ) ∧ [ x ∈ X { y | P xy } = c ] , (13) ( ψ ∧ χ ) F ≡ ψ F ∧ χ F , (14) ( ¬ ψ ) F ≡¬ ψ F , (15) ( ∀ xψ ) F ≡ ✷ ∀ x ( N ( x ) → ψ F ) , (16) ( ∀ X n ψ ) F ≡ ✷ ∀ X n ( ∀ x , . . . , x n ( X n x . . . x n → N ( x ) ∧ · · · ∧ N ( x n )) → ψ F ) . (17)To see that this is a generalised translation all that remains to be shown is that deductionis preserved by our translation. We need this result for both E PI and I PI . Lemma . Let ϕ , . . . , ϕ n , ψ be unnested formulas in the language of PA with free vari-ables v , . . . , v m , it follows that if ϕ , . . . , ϕ n ⊢ ψ , then N ( v ) , . . . , N ( v m ) , ϕ F , . . . , ϕ F n (cid:15) PI ψ F .Further, it is PA -provable that if ϕ , . . . , ϕ n ⊢ ψ then ACA ⊢ “ N ( v ) , . . . , N ( v m ) , ϕ F , . . . ,ϕ F n (cid:15) PI ψ F ”. The first part of this Lemma is similar to Linnebo (2013, Thm. 5.4.). But he proves aversion of this which does not restrict the quantifiers to a domain. The modification to ourcase is simple and so we omit the proof.On its own a translation is not very interesting. However, a translation is an interpretation if the translations of the axioms of the interpreted theory can be proven in the interpretingtheory.
Definition . Let T A and T B be L A and L B theories respectively, where a theory isa set of sentences not necessarily closed under deduction. A generalised translation ( · ) G : L A → L B interprets T A in T B , if for all L A unnested sentences χ : T A ⊢ L A χ ⇒ T B ⊢ L B χ G (18)It is a recursive interpretation if the collection of L A and L B formulas are recursive, T A and T B are also recursive, as is ( · ) G , and there are recursive maps from proofs to proofswhich witness the truth of equations (9) and (18). If T extends PA , then say that theinterpretation is T -verifiable if the recursive functions are provably total in T and if theuniversal closures of the arithmetized versions of 9 and 18 are provable in T .So, the proofs of Sections 3 and 4 show our translation is an interpretation of PA in E PI .However, to show it is an interpretation in I PI a certain level of caution is needed because I PI does not have a background derivability relation. To resolve this, we take ϕ , . . . , ϕ n ⊢ L PI ϕ to be ACA ⊢ “ ϕ , . . . , ϕ n (cid:15) PI ϕ ”, where this is as defined in Appendix B. And, of course I PI Recall that we formalised I PI in ACA , and those interested in the nuts and bolts are directed toAppendix B. HE POTENTIAL IN FREGE’S THEOREM ϕ such that ACA ⊢ “ (cid:15) PI ϕ ”.We then need to show the following: Lemma . For all sentences ϕ in the language of PA , if PA ⊢ ϕ then ACA ⊢ “ (cid:15) PI ϕ F ” . Further, it is PA -provable that if PA ⊢ ϕ then ACA ⊢ “ (cid:15) PI ϕ F ” . Proof.
By Lemmas 3.2-3.6 and 4.2 and 4.3 we know that if ϕ is an axiom of PA then ACA ⊢ “ (cid:15) PI ϕ F ”. Assume PA ⊢ ϕ not an axiom, then there are n axioms of PA , ϕ , . . . , ϕ n , such that ϕ , . . . , ϕ n ⊢ ϕ . Then as we can always take the universal closure ofaxioms and ϕ is a sentence it follows by Lemma 5.2 that ACA ⊢ “ ϕ F , . . . , ϕ F n (cid:15) PI ϕ F ”.Given that the axioms are PI valid, it follows that ACA ⊢ “ (cid:15) PI ϕ F ”. ⊣ This final piece gives us the proof of:
Theorem . There is a generalised translation from the language of PA to thesecond-order modal language with octothorpe that interprets PA in I PI . Further, this isa PA -verifiable generalised interpretation. To prove the first half of Theorem 1.8 we need to define formulas that pick out the numbersin PA and E PI . In PA let τ ( x ) ≡ ( x = 0) and τ n +1 ( x ) ≡ ∃ y ( τ n ( y ) ∧ Syx ). In E PI let σ ( x ) ≡ ( x = 0) and σ n +1 ( x ) ≡ ✸ ∃ y ∈ N ( σ n ( y ) ∧ Syx ). Note that ( τ ( x )) F ≡ ( x = 0) F ≡ σ ( x ) and ( τ n +1 ( x )) F ≡ ( ∃ y ( τ n ( y ) ∧ Syx )) F ≡ ✸ ∃ y ∈ N (( τ n ( y )) F ∧ Syx ) ≡ σ n +1 ( x ). Withthis we can state the following preliminary Lemma; we omit the proof which is long but notilluminating: Lemma . For every k ≥ and every unnested formula θ ( x , . . . , x k ) in the signatureof PA and every k -tuple of natural numbers n , . . . , n k one has that : N | = θ ( n , . . . , n k ) = ⇒ (cid:15) PI ∀ x , . . . , x k ∈ N ( k ^ i =1 σ n i ( x i ) → θ F ( x , . . . , x k ))(19) In the case of k = 0 , this is to say: for every unnested sentence θ in the signature of PA one has that N | = θ = ⇒ (cid:15) PI θ F (20)Theorem 1.8.i follows from (20) of Lemma 5.5. This give us our proof of: Theorem . There is a generalised translation from the language of PA to the second-order modal language with octothorpe that interprets TA in E PI . §
6. Proof of Theorem 1.9.
It has been shown by Linnebo and Shapiro (2019, §
7) thatthe Linnebo translation cannot interpret comprehension because modalized comprehensionrequires the existences of a set of all possibly existing things. However, this leaves open thequestion of whether there is a different translation which can interpret PA . Here we willdemonstrate that there is no translation from TA to E PI nor from PA to I PI by provingTheorem 1.9, our second main theorem. The first part of Theorem 1.9 follows from relativelysimple Tarskian considerations: Theorem . There is no generalised translation from the language of PA to thesecond-order modal language with octothorpe that interprets TA in E PI . Proof.
Assume for a contradiction that there is an interpretation ( · ) G that interprets TA in E PI . Note that as TA is complete it follows that this is a faithful interpretation;i.e. if (cid:15) PI ϕ G then N (cid:15) ϕ . As E PI is Π -definable it follows that there is a predicate P such8 WILL STAFFORD that for all ϕ in the second-order modal language with octothorpe we have (cid:15) PI ϕ if and onlyif N (cid:15) P (“ ϕ ”). (Here we use quotation marks for G¨odel numbering for both the languageof PA and the second-order modal language with octothorpe.) But then as generalisedtranslations are recursive we can represent ( · ) G in N as g . It follows that P ( g (“ ψ ”)), where ψ is in the language of PA , is a truth predicate for TA . But this contradicts Tarski’stheorem. ⊣ The proof of the second part of the theorem is trickier and requires G¨odelian consider-ations. Recall the definition of T -verifiable generalised translation and interpretation fromDefinitions 5.1 and 5.3 in Section 5. There we proved that we have a PA -verifiable interpre-tation of PA in I PI by Lemma 5.4. Given that we defined I PI ⊢ ϕ as ACA ⊢ “ (cid:15) PI ϕ ”, that is PA ⊢ ∀ ϕ [“ PA ⊢ ϕ ” → “ ACA ⊢ “ (cid:15) PI ϕ F ””]. Here we show that there is no PA -verifiableinterpretation of PA in I PI . We can write this as: there is no generalised translation ( · ) G from the language of PA to the second-order modal language with octothorpe such that PA ⊢ ∀ ϕ [“ PA ⊢ ϕ ” → “ ACA ⊢ “ (cid:15) PI ϕ G ””]. Theorem . There is no generalised translation from the language of PA to thesecond-order modal language with octothorpe that PA -verifiably interprets PA in I PI . Proof.
The systems Π k - CA are subsystems of PA that have comprehension for Π k formulas. As proofs are finite and so can only use finitely many instances of the compre-hension schema any interpretation which is PA -verifiable will also be Π k - CA -verifiable forsome k ≥
1. Let ϕ , . . . , ϕ n be a finite axiomatisation of Π k - CA for some k ≥ k - CA -verifiabletranslation ( · ) G from the languge of PA to the second-order modal language with octothorpethat interprets PA in I PI , that Π k - CA proves its own consistency. This contradicts G¨odel’ssecond incompleteness theorem and so shows that no such ( · ) G can exist.Note that PA ⊢ ϕ , . . . , ϕ n as all Π k - CA are subsystems of PA . We are assuming that( · ) G interprets PA in I PI , so it follows that ACA ⊢ “ (cid:15) PI ϕ G , . . . , ϕ G n ”. Let A be a model ofΠ k - CA for some k . So, we have A (cid:15) “ (cid:15) PI ϕ G , . . . , ϕ G n ”. If M is the minimal model fromExample 1.4 relative to A then we have then we have A (cid:15) “ M (cid:15) ϕ G , . . . , ϕ G n ”.Now we show that A (cid:15) ¬ P rv ϕ ,...,ϕ n ( ψ ∧ ¬ ψ ), that is the consistency of Π k - CA . Assumefor a contradiction that A (cid:15) ∃ πP rf ϕ ,...,ϕ n ( π, ψ ∧ ¬ ψ ). Then as ( · ) G is a Π k - CA -verifiableinterpretation it follows A (cid:15) P rf
ACA ( π G , “ ϕ G , . . . , ϕ G n (cid:15) PI ψ G ∧ ¬ ψ G′′ ).Recall that Π - CA proves Σ -reflection for ACA (cf. Simpson (2009) Theorem VII.6.9.(4)p. 298 and Theorem VII.7.6.(1) p. 305). As Π - CA ⊆ Π k - CA , this means that for any Π statement ψ we know Π k - CA proves P rv
ACA ( ψ ) → ψ . For all ψ , we know that “ (cid:15) PI ψ ”is Π and similarly for the local derivability relation (see Appendix B). It follows that A (cid:15) “ ϕ G , . . . , ϕ G n (cid:15) PI ψ G ∧ ¬ ψ G ” and as A (cid:15) “ M (cid:15) ϕ G , . . . , ϕ G n ”. It follows that A (cid:15) “ M (cid:15) ψ G ∧ ¬ ψ G ”. And so A (cid:15) “ M | = ψ G ” and A (cid:15) “ M | = ¬ ( ψ G )”. ⊣ We have now shown the two main results set out in the introduction. §
7. Conclusion.
We started with the worry that Hume’s Principle had only infinitemodels and so any claim that it was analytic would mean that the claim that there areinfinitely many objects is analytic. This worry has been noted before in the literature onneo-logicism, but little has been done to address it. Hale and Wright (2001) state thatwithout this the neo-logicist project cannot even get off the ground:To require of an acceptable abstraction that it should not be (even) weakly infla-tionary [that is require a countable infinity] would stop the neo-Fregean project
HE POTENTIAL IN FREGE’S THEOREM § Appendix A. Formal Theories.
Here we will spell out the theories other than E PI and I PI which are used in the proofs above. Unlike E PI and I PI none of these are modaltheories, however, most are second-order theories.The weakest theory we consider is first-order Robinson’s Q . For a more complete referencesee, for example, H´ajek and Pudl´ak (1998, p. 28). Definition
A.1 . Q is the usual formalization of Robinson’s arithmetic. It consists of theuniversal closure of the following axioms: s ( x ) = 0; (Q1) s ( y ) = s ( z ) → y = z ;(Q2) x + 0 = x ; (Q3) x + s ( y ) = s ( x + y );(Q4) x × x × s ( y ) = ( x × y ) + y. (Q6)Note that in the body of the text we do not use this formulation but rather one withrelations instead than functions. We have offered this formulation for readability. Therelation formulation gives you the obvious translation of the above, plus an additional 6axioms ensuring that the relations S, + , × are the graphs of functions.We also consider the extensions of Q to PA by the addition of the first-order inductionschema, and PA by the addition of the second-order induction axiom and ComprehensionSchema. PA is a first-order theory, but PA is a second-order theory. We use a capital S for the relational successor and lower case s for the functional. WILL STAFFORD
Definition
A.2 . PA is Q plus the induction schema, where ϕ is a first-order formula:( ϕ (0) ∧ ∀ x ( ϕx → ϕ ( s ( x )))) → ∀ xϕ ( x )(Induction Schema (IS)) PA is Q plus the induction axiom and Comprehension Schema: ∀ P [( P ∧ ∀ x ( P x → P ( s ( x )))) → ∀ xP x ](Induction Axiom (IS)) ∀ ¯ y, ¯ Y ∃ X ∀ x ( X ( x ) ↔ ϕ ( x, ¯ y, ¯ Y ))(Comprehension Schema (CS))In the Comprehension Schema ϕ can be any formula of the language of PA in which X does not occur free.Again in the body of the text we use the natural adaptation to the setting of relations ratherthan functions. There are also two theories we use that are second-order and between PA and PA in strength. They both restrict comprehension. So, we first need to define theformulas we restrict to: Definition
A.3 . (Simpson, 2009, I.3.1, p. 6) An Arithmetical formula is a formula in thelanguage of PA which does not contain any set quantifiers, though it may contain free setvariables.With this we can state ACA : Definition
A.4 . (Simpson, 2009, I.3.2, p. 7) ACA is Q plus the Induction Axiom andArithmetical Comprehension: ∀ ¯ y, ¯ Y ∃ X ∀ x ( X ( x ) ↔ ϕ ( x, ¯ y, ¯ Y ))(Arithmetical Comprehension Schema (ACS))Where ϕ has to be an arithmetical formula and X may not occur free.Note that as every formula of PA is arithmetical, and ACA contains the second-orderinduction axiom, every instance of the first-order induction schema is provable in ACA .The next theories of arithmetic to be considered here are the Π k - CA which are usedin the proof of Theorem 1.9. To define this theory, we first need to define Π k (and Σ k )formulas: Definition
A.5 . (Simpson, 2009, I.5.1, p. 16) A Π formula is a formula in the languageof PA of the form ∀ X , . . . , X n ϕ where X , . . . , X n are set variables and ϕ is an arithmeticalformula.A Σ formula is a formula in the language of PA of the form ∃ X , . . . , X n ϕ where X , . . . , X n are set variables and ϕ is an arithmetical formula.A Π k formula is a formula in the language of PA of the form ∀ X , . . . , X n ϕ where X , . . . , X n are set variables and ϕ is a Σ k − formula.A Σ k formula is a formula in the language of PA of the form ∃ X , . . . , X n ϕ where X , . . . , X n are set variables and ϕ is Π k − formula.The definition of Π k - CA is much like the definition of ACA , except that the restriction onthe comprehension axiom is broadened to include all Π k formulas: Definition
A.6 . (Simpson, 2009, I.5.2, p. 17) Π k - CA is Q plus the Induction Axiomand Π k Comprehension: ∀ ¯ y, ¯ Y ∃ X ∀ x ( X ( x ) ↔ ϕ ( x, ¯ y, ¯ Y ))(Π k Comprehension Schema (Π k CS))Where ϕ has to be a Π k formula and X may not occur free. HE POTENTIAL IN FREGE’S THEOREM N be { ω, , s, + , ×} where eachterm is interpreted as it is in the metatheory and N be N with P ( ω n ) as the domain of thesecond-order quantifiers. N is the intended model of Q and PA , while N is the intendedmodel of PA , ACA , and Π k - CA for all k . As is well known, by G¨odel’s incompletenesstheorems none of the theories we have seen so far are complete. We can define the completetheories of these models: Definition
A.7 . Let TA be { ϕ | N (cid:15) ϕ } and TA be { ϕ | N (cid:15) ϕ } .For the sake of completeness, we here define Hume’s Principle ( HP ). This system issecond-order also and consists of the cardinality principle displayed in Equation HP onpage 1, the full Comprehension Schema, as in PA , and full comprehension for binary rela-tions: ∀ ¯ y, ¯ Y ∃ X ∀ x, z ( X ( x, z ) ↔ ϕ ( x, z, ¯ y, ¯ Y ))(Binary Comprehension Schema (BCS))Comprehension for binary relations is required because the definition of HP quantifies overbijections and when spelt out fully this turns out to be the claim that there is a second-orderbinary relation which is the graph of a bijection between the two sets. § Appendix B. Formal definition of I PI . In the introduction we gave I PI as the set { ϕ | ACA ⊢ ‘ (cid:15) PI ϕ ’ } . Here we will layout explicitly what we mean by defining the arithmetizationof (cid:15) PI in ACA .It is importaint to note that the second-order variables in I PI are taken to first-ordervariables in ACA . If all the first-order variables of I PI are of the form x i and all the second-order variables of I PI are of the form Y j then let all the first-order variables of ACA be ofthe form x i and Y j , and the second-order variables of ACA be of the form Z v . In practicewe will not stick to this strict distinction, but it can always be implemented by renamingthe variables.We do not restrict the domain of the first-order variables of I PI ; there is no need to pickout a subset of the domain of a model of ACA . However, the second-order variables of I PI need to be restricted to codes for finite sets of numbers ordered by strict less than. This isn’tdifficult, we can simply borrow the coding found in the proof of incompleteness. A morecomplete explication can be found in Simpson (2009, Ch. 2.2). The second-order variablesare required to be to some sequence π (0) n + · · · + π ( m ) n m where π ( i ) gives the i th primeand n < n < · · · < n m . Let Seq ( Y ) be the name of the relation that ensures Y has theabove properties. Further, let nSeq ( Y ) mean that Y codes n -tuples of numbers. We willuse this to code relations and relational variables. If x is the number of a sequence then let[ x ] i be the i th element and ln ( x ) is the length of x .We want to code PI models as sets of natural numbers. We know that we can alwayscombine countably many countably infinite sets (just code n a member of the i th set as2 i + 3 n ). As such we will just show how to code W, R, D, , a as separate sets of naturalnumbers. Further, with R, D, , a we will talk about pairs ( x, y ), this should be understoodas standing for the code 2 x + 3 y .(B.1) Let W be infinite ( ∀ x ∈ W ∃ y ∈ W ( y > x )), (B.2) let R be such that(a) for all ( i, j ) ∈ R we have that i, j ∈ W , Recall that our definition demanded that our set of worlds be countable. We cannot capture this in
ACA in the sense that ACA has none standard models but we will have that we do not have more worldsthan ACA thinks there are natural numbers, which is sufficent for the role this plays in the proofs. WILL STAFFORD (b) ∀ x ∈ W R ( x, x ) (reflexive),(c) ∀ x, y, z ∈ W ( R ( x, y ) ∧ R ( y, z ) → R ( x, z )) (transitive),(d) ∀ x, y ∈ W ( R ( x, y ) ∧ R ( y, x ) → x = y ) (anti-symmetric),(e) ∀ x, y ∈ W ∃ z ∈ W ( R ( x, z ) ∧ R ( y, z )) (directed),(B.3) let D be such that(a) D ( w, Y ) implies that w ∈ W and Seq ( Y ),(b) ∀ w ∈ W ∃ Y ∈ Seq ( D ( w, Y ) ∧ ln ( Y ) >
0) (every world has at least one element),(c) D is the graph of a function from W to Seq ,(d) if R ( i, j ) and i = j and D ( i, X ) and D ( j, Y ) then ∃ u ∀ v ([ X ] v = [ Y ] u ) (there issomething in Y not in X ) and ∀ v < ln ( X ) ∃ u ([ X ] v = [ Y ] u ) (everything in X is in Y ),(B.4) let a be such that for each n there is exactly one x such that a ( n, x ) and if a ( n, x ) and a ( m, x ) then n = m , we then define Y, x ) as
Seq ( Y ) ∧ a ( ln ( Y ) , x ).Given a set of numbers M we will write M ∈
P IM to signify the set meets (B.1)–(B.4).We define sb (subset) as follows Y ∈ sb ( X ) iff Seq ( Y ) ∧ ∀ i < ln ( Y ) ∃ j ([ X ] j = [ Y ] i ). Indefining the arithmetisation note that we add free-variables for the model and the world,we will use W M , R M , D M , M , but these can be defined in terms of the model. So, if ϕ is a formula in the modal second-order language with octothorpe we translate it to some ψ ( w, W M , R M , D M , M ) in the language of arithmetic. We define the arithmetisation asfollows:( x i = x j ) ∗ ≡ x i = x j (21) ( x i = Y j ) ∗ ≡ M ( Y j , x i )(22) ( Y j x i ) ∗ ≡∃ u ( x i = [ Y j ] u )(23) ( ∀ xϕ ) ∗ ≡∀ x ( ∃ Y ∈ Seq ( D M ( w, Y ) ∧ ∃ u ( x = [ Y ] u )) → ( ϕ ) ∗ )(24) ( ∀ Y ϕ ) ∗ ≡∀ Y ∈ Seq ( ∃ X ∈ Seq ( D M ( w, X ) ∧ Y ∈ sb ( X )) → ( ϕ ) ∗ )(25) ( ∀ P n ϕ ) ∗ ≡∀ P n ∈ nSeq (26) ( ∃ X ∈ Seq ( D M ( w, X ) ∧ ∀ ( x , . . . , x n ) ∈ P n ( ^ ≤ i ≤ n ∃ j [ X ] j = x i )) → ( ϕ ) ∗ )( ✷ ϕ ) ∗ ≡∀ s ∈ W M ( R M ( w, s ) → ( ϕ ) ∗ [ w/s ])(27)where we commute over the logical connectives. This means that every formula arithmetisedis arithmetical as defined in Appendix A. For example, ✷ ∀ v ✸ ∃ Z ( v = Z ) becomes(28) ∀ s ∈ W M ( R M ( w, s ) → ∀ v ( ∃ Y ( D M ( s, Y ) ∧ ∃ u ( v = [ Y ] u ) →∃ s ′ ∈ W M ( R M ( s, s ′ ) ∧ ∃ Z ∈ Seq ( ∃ X ( D M ( w, X ) ∧ Z ∈ sb ( X ) ∧ M ( Z, v ))))) . Note ‘ (cid:15) PI ϕ ’ means ∀ M ∈ P IM ∀ w ∈ W M ( ϕ ) ∗ . It follows that this is then a Π formula.Hence, if one were proceeding very formally, we would define I PI as the set of all the ϕ suchthat ACA ⊢ ∀ M ∈ P IM ∀ w ∈ W M ( ϕ ) ∗ . References.
Bell, John L. (1999). “Frege’s Theorem in a Constructive Setting”. In:
Journal of SymbolicLogic
Philosophical Review
Logic, Logic, and Logic . Cambridge, MA: Harvard University Press.
EFERENCES
Fixing Frege . Princeton: Princeton University Press.Button, Tim and Sean Walsh (2018).
Philosophy and Model Theory . Oxford: Oxford Uni-versity Press.Cook, Roy T (2007). “Introduction”. In:
The Arch´e Papers on the Mathematics of Abstrac-tion . Ed. by Roy T Cook. Vol. 71. The Western Ontario Series in Philosophy of Science.Berlin: Springer, pp. xv–xxxvii.Demopoulos, William (June 1994). “Frege and the rigorization of analysis”. In:
J. Philos.Logic
The Oxford Handbook of Philosophy of Math-ematics and Logic . Ed. by Stewart Shapiro. Oxford: Oxford University Press, pp. 590–624.Fitting, Melvin and Richard L. Mendelsohn (1998).
First-Order Modal Logic . Dordrecht:Kluwer Academic Publishers.Frege, Gottlob (1884).
Die Grundlagen der Arithmetik: eine logisch mathematische Un-tersuchung ¨uber den Begriff der Zahl . (translated as
The Foundations of Arithmetic: Alogico-mathematical enquiry into the concept of number , by J.L. Austin, Oxford: Blackwell,second revised edition, 1974.) Breslau: W. Koebner.– (1893).
Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet . (translation as
BasicLaws of Arithmetic: Derived using concept-script by P. Ebert and M. Rossberg (with C.Wright), Oxford: Oxford University Press, 2013.) Jena: Verlag Hermann Pohle.H´ajek, Petr and Pavel Pudl´ak (1998).
Metamathematics of First-Order Arithmetic . Perspec-tives in Mathematical Logic. Berlin: Springer.Hale, Bob and Crispin Wright (2001).
The Reason’s Proper Study . Oxford: Oxford UniversityPress.Heck, Richard Kimberly (1993). “The development of arithmetic in Frege’s Grundgesetzeder Arithmetik”. In:
Journal of Symbolic Logic
From Frege to G¨odel : A Source Book in Mathematical Logic,1879-1931 . Cambridge: Harvard University Press.Hodes, Harold (1984). “Logicism and the Ontological Commitments of Arithmetic”. In:
TheJournal of Philosophy
Synthese
Model Theory . Cambridge: Cambridge University Press.Kim, Joongol (2015). “A Logical Foundation of Arithmetic”. In:
Studia Logica
Journal ofPhilosophical Logic
The Review of SymbolicLogic
Thin objects: An abstractionist account . Oxford: Oxford University Press.Linnebo, Øystein and Stewart Shapiro (Mar. 2019). “Actual and Potential Infinity”. In:
Noˆus
MathematicalLogic Quarterly
Mathematics in Philosophy: Selected Es-says . Ithaca: Cornell University Press, pp. 298–341.Putnam, Hilary (1967). “Mathematics Without Foundations”. In:
Journal of Philosophy
REFERENCES
Shapiro, Stewart and Øystein Linnebo (2015). “Frege Meets Brouwer”. In:
Review of Sym-bolic Logic
Subsystems of Second Order Arithmetic . Perspectives in Logic. Cam-bridge: Cambridge University Press.Stanley, Jason (1997). “Names and Rigid Designation”. In:
A Companion to the Philosophyof Language . Ed. by Bob Hale and Crispin Wright. Malden: Blackwell, pp. 555–585.Studd, JP (2016). “Abstraction Reconceived”. In:
The British Journal for the Philosophyof Science
AXIOMS
Bull. Symb. Log.
Principia Mathematica . Vol. 1. Cambridge:Cambridge University Press.Williamson, Timothy (2013).
Modal Logic as Metaphysics . en. Oxford: Oxford UniversityPress.
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