TThe Strength of Abstraction with PredicativeComprehension
Sean Walsh ∗ November 16, 2015
Abstract
Frege’s theorem says that second-order Peano arithmetic is interpretable in Hume’sPrinciple and full impredicative comprehension. Hume’s Principle is one example of an abstraction principle , while another paradigmatic example is Basic Law V from Frege’s
Grundgesetze . In this paper we study the strength of abstraction principles in thepresence of predicative restrictions on the comprehension schema, and in particularwe study a predicative Fregean theory which contains all the abstraction principleswhose underlying equivalence relations can be proven to be equivalence relations in aweak background second-order logic. We show that this predicative Fregean theoryinterprets second-order Peano arithmetic (cf. Theorem 3.2).
Contents ∗ Department of Logic and Philosophy of Science, 5100 Social Science Plaza, University of California,Irvine, Irvine, CA 92697-5100, U.S.A., [email protected] or [email protected] a r X i v : . [ m a t h . L O ] N ov Introduction
The main result of this paper is a predicative analogue of Frege’s Theorem (cf. Theorem 3.2).Roughly, Frege’s theorem says that one can recover all of second-order Peano arithmetic usingonly the resources of Hume’s Principle and second-order logic. This result was adumbratedin Frege’s
Grundlagen of 1884 ([10], [13]) and the contemporary interest in this result is dueto Wright’s 1983 book
Frege’s Conception of Numbers as Objects ([34]). For more on thehistory of this theorem, see the careful discussion and references in Heck [20] pp. 4-6 andBeth [1].More formally, Frege’s theorem says that second-order Peano arithmetic is interpretablein second-order logic plus the following axiom, wherein the cardinality operator
Hume’s Principle : ∀ X, Y ( X = Y ↔ ∃ bijection f : X → Y )Of course, one theory is said to be interpretable in another when the primitives of theinterpreted theory can be defined in terms of the resources of the interpreting theory so thatthe translations of theorems of the interpreted theory are theorems of the interpreting theory(cf. [30] § § § ϕ ( x ) in one free first-order variable determines a second-order entity:(1.2) ∃ F ∀ x ( F x ↔ ϕ ( x ))The traditional proof of Frege’s Theorem uses instances of this comprehension schema inwhich some of the formulas in question contain higher-order quantifiers (cf. [29] p. 1690equations (44)-(45)). However, there is a long tradition of predicative mathematics , in whichone attempts to ascertain how much one can accomplish without directly appealing to suchinstances of the comprehension schema. This was the perspective of Weyl’s great book DasKontinuum ([33]) and has been further developed in the work of Feferman ([7], [8]). Manyof us today learn and know of this tradition due to its close relation to the system
ACA ofFriedman and Simpson’s project of reverse mathematics ([15], [26]).However, outside of the inherent interest in predicative mathematics, considerations re-lated to Frege’s philosophy of mathematics likewise suggest adopting the predicative per-spective. For, Wright and Hale ([18], cf. [4]) have emphasized that Hume’s Principle (1.1)is a special instance of the following:(1.3) A[E] : ∀ X, Y ( ∂ E ( X ) = ∂ E ( Y ) ↔ E ( X, Y ))wherein E ( X, Y ) is a formula of second-order logic and ∂ E is a type-lowering operator takingsecond-order entities and returning first-order entities. These principles were called abstrac-tion principles by Wright and Hale, who pointed out that the following crucial fifth axiomof Frege’s Grundgesetze of 1893 and 1903 ([11], [14]) was also an abstraction principle:(1.4)
Basic Law V : ∀ X, Y ( ∂ ( X ) = ∂ ( Y ) ↔ X = Y )2he operator ∂ as governed by Basic Law V is called the extension operator and the first-order entities in its range are called extensions . Regrettably, there is no standard notationfor the extension operator, and so some authors write § X in lieu of ∂ ( X ). In what follows,the symbol ∂ without any subscripts will be reserved for the extension operator, whereas thesubscripted symbols ∂ E will serve as the notation for the type-lowering operators present inarbitrary abstraction principles (1.3).While the Russell paradox shows that Basic Law V is inconsistent with the full compre-hension schema (1.2) (cf. [29] p. 1682), nevertheless Basic Law V is consistent with predica-tive restrictions, as was shown by Parsons ([25]), Heck ([19]), and Ferreira-Wehmeier ([9]).This thus suggests the project of understanding whether there is a version of Frege’s theo-rem centered around the consistent predicative fragments of the Grundgesetze . This projecthas been pursued in the last decades by many authors such as Heck ([19]), Ganea ([16]),and Visser ([28]). Their results concerned the restriction of the comprehension schema (1.2)to the case where no higher-order quantifiers are permitted. One result from this body ofwork says that Basic Law V (1.4) coupled with this restriction on the comprehension schemais mutually interpretable with Robinson’s Q . Roughly, Robinson’s Q is the fragment offirst-order Peano arithmetic obtained by removing all the induction axioms. (For a precisedefinition of Robinson’s Q , see [17] p. 28, [26] p. 4, [29] p. 1680, [30] p. 106). Additionalwork by Visser allows for further rounds of comprehension and results in systems mutuallyinterpretable with Robinson’s Q plus iterations of the consistency statement for this theory,which are likewise known to be interpretable in other weak arithmetics ([28] p. 147). In his2005 book ([3]), Burgess surveys these kinds of developments, and writes:[. . . ] I believe that no one working in the area seriously expects to get verymuch further in the sequence Q m while working in predicative Fregean theoriesof whatever kind ([3] p. 145).Here Q m is the expansion of Robinson’s Q by finitely many primitive recursive functionsymbols and their defining equations along with induction for bounded formulas ([3] pp.60-63), so that Burgess records the prediction that predicative Fregean theories will beinterpretable in weak arithmetics.The main result of this paper suggests that this prediction was wrong, and that predica-tive Fregean theories can interpret strong theories of arithmetic (cf. Theorem 3.2). While weturn presently to developing the definitions needed to precisely state this result, let us sayby way of anticipation that part of the idea is to work both with (i) an expanded notion ofa “Fregean theory,” so that it includes several abstraction principles, such as Basic Law V,in addition to Hume’s Principle, and (ii) an expanded notion of “predicativity,” in whichone allows some controlled instances of higher-order quantifiers within the comprehensionschema (1.2). Hence, of course, it might be that Burgess and others had merely conjecturedthat predicative Fregean theories in a more limited sense were comparatively weak.This paper is part of a series of three papers, the other two being [31] and [32]. Thesepapers collectively constitute a sequel to our paper [29], particularly as it concerns themethods and components related to Basic Law V. In that earlier paper, we showed thatHume’s Principle (1.1) with predicative comprehension did not interpret second-order Peano3rithmetic with predicative comprehension (cf. [29] p. 1704). Hence at the outset of thatpaper, we said that “in this specific sense there is no predicative version of Frege’s Theorem”([29] p. 1679). The main result of this present paper (cf. Theorem 3.2) is that when weenlarge the theory to a more inclusive class of abstraction principles containing Basic Law V,we do in fact succeed in recovering arithmetic.This paper depends on [31] only in that the consistency of the predicative Fregean theorywhich we study here was established in that earlier paper (cf. discussion at close of nextsection). In the paper [32], we focus on embedding the system of the Grundgesetze into asystem of intensional logic. The alternative perspective of [32] then suggests viewing theconsistent fragments of the
Grundgesetze as a species of intensional logic, as opposed to aninstance of an abstraction principle.This paper is organized as follows. In § § § The predicative Fregean theory with which we work in this paper is developed within theframework of second-order logic. The language L of the background second-order logicis an ω -sorted system with sorts for first-order entities, unary second-order entities, binarysecond-order entities etc. Further, following the Fregean tradition, the first-order entities arecalled objects , the unary second-order entities are called concepts , and the n -ary second-orderentities for n ≥ n -ary concepts . Rather than introduce any primitive notationfor the different sorts, we rather employ the convention of using distinctive variables for eachsort: objects are written with lower-case Roman letters x, y, z, a, b, c . . . , concepts are writtenwith upper-case Roman letters X, Y, Z, A, B, C, F, G, H, U, . . . , n -ary concepts for n > R, S, T , and n -ary concepts are written with theRoman letters f, g, h when they are graphs of functions.Besides the sorts, the other basic primitive of the signature of the background second-order logic L are the predication relations. One writes Xa to indicate that object a hasproperty or concept X . Likewise, there are predication relations for n -ary concepts, whichwe write as R ( a , . . . , a n ). The final element of the signature L of the background second-order logic are the projection symbols. The basic idea is that one wants, primitive in thesignature L , a way to move from the binary concept R and the object a to its projec-tion R [ a ] = { b : R ( a, b ) } . We assume that the signature L of the background second-orderlogic is equipped with symbols ( R, a , . . . , a m ) (cid:55)→ R [ a , . . . , a m ] from ( m + n )-ary concepts R and an m -tuple of objects ( a , . . . , a m ) to an n -ary concept R [ a , . . . , a m ] = { ( b , . . . , b n ) : R ( a , . . . , a m , b , . . . , b n ) } . Further, typically in what follows we avail ourselves of the tuplenotation a = a , . . . , a n and thus write predication and projection more succinctly as R ( a )4nd R [ a ], respectively.All this in place, we can then formally define the signature L of the background second-order logic as follows: Definition 2.1.
The signature L of the background second-order logic is a many-sortedsignature which contains (i) a sort for objects and for each n ≥ a sort for n -ary concepts,(ii) for each n ≥ , an ( n + 1) -ary predication relation symbol R ( a , . . . , a n ) which holdsbetween an n -ary concept R and an n -tuple of objects a , . . . , a n , and (iii) for each n, m ≥ ,an ( m + 1) -ary projection function symbol ( R, a , . . . , a m ) (cid:55)→ R [ a , . . . , a m ] from an ( m + n ) -ary concept R and an m -tuple of objects ( a , . . . , a m ) to an n -ary concept R [ a , . . . , a m ] . As is usual in many-sorted signatures, we adopt the convention that each sort has its ownidentity symbol, so that technically cross-sortal identities are not well-formed. But we con-tinue to write all identities with the usual symbol “=” for the ease of readability.The expansions of second-order logic with which we work are designed to handle abstrac-tion principles (1.3). Hence, suppose that L is an expansion of L . Suppose that E ( R, S ) isan L -formula with two free n -ary relation variables for some n ≥
1, with all free variables of E ( R, S ) explicitly displayed. Then we may expand L to a signature L [ ∂ E ] which contains anew function symbol ∂ E which takes n -ary concepts R and returns the object ∂ E ( R ). Thenthe following axiom, called the abstraction principle associated to E , is an L [ ∂ E ]-sentence:(2.1) A[E] : ∀ R, S ( ∂ E ( R ) = ∂ E ( S ) ↔ E ( R, S ))This generalizes the notion of an abstraction principle (1.3) described in the previous sectionin that the domain of the operator ∂ E can be n -ary concepts for any specific n ≥ R be a binary concept and let Field( R ) be the unary concept F such that F x iff there is a y such that Rxy or Ryx . Then consider the following formula E ( R, S ) on binary concepts:[(Field( R ) , R ) | = wo ∨ (Field( S ) , S ) | = wo] → (2.2) ∃ isomorphism f : (Field( R ) , R ) → (Field( S ) , S )In this, “wo” denotes the natural sentence in the signature of second-order logic whichsays that a binary concept is a well-order, i.e. a linear order such that every non-emptysubconcept of its domain has a least element. It’s not too difficult to see that E ( R, S ) is anequivalence relation on binary concepts, and that two well-orders will be E -equivalent if andonly if they are order-isomorphic. Just as the Russell paradox shows that Basic Law V (1.4)is inconsistent with the full comprehension schema, so one can use the Burali-Forti paradoxto show that A [ E ] for this E in equation (2.2) is inconsistent with the full comprehensionschema (cf. [21] p. 138 footnote, [2] pp. 214, 311). To handle these abstraction principleswe need to adopt restrictions on the comprehension schema, to which we presently turn.There are three traditional predicative varieties of the comprehension schema: the first-order comprehension schema, the ∆ -comprehension schema, and the Σ -choice schema (cf.[26] VII.5-6, [29] Definition 5 p. 1683). However, to make the comparison with the fullcomprehension schema (1.2) precise, we should restate it to include not only concepts but n -ary concepts for all n ≥ efinition 2.2. Suppose that L is an expansion of L . Then the Full Comprehension Schemafor L -formulas consists of all axioms of the form ∃ R ∀ a ( Ra ↔ ϕ ( a )) , wherein ϕ ( x ) isan L -formula, perhaps with parameters, and x abbreviates ( x , . . . , x n ) and R is an n -aryconcept variable for n ≥ that does not appear free in ϕ ( x ) . The most restrictive predicative version of the comprehension schema is then the following,where the idea is that no higher-order quantifiers are allowed in the formulas: Definition 2.3.
Suppose that L is an expansion of L . The First-Order ComprehensionSchema for L -formulas consists of all axioms of the form ∃ R ∀ a ( Ra ↔ ϕ ( a )) , wherein ϕ ( x ) is an L -formula with no second-order quantifiers but perhaps with parameters, and x abbre-viates ( x , . . . , x n ) and R is an n -ary concept variable for n ≥ that does not appear freein ϕ ( x ) . A more liberal version of the comprehension schema is the so-called ∆ -comprehensionschema. A Σ -formula (resp. Π -formula) is one which begins with a block of existentialquantifiers (resp. universal quantifiers) over n -ary concepts for various n ≥ Definition 2.4.
Suppose that L is an expansion of L . Then the ∆ -Comprehension Schemafor L -formulas consists of all axioms of the form (2.3) ( ∀ x ϕ ( x ) ↔ ψ ( x )) → ∃ R ∀ a ( Ra ↔ ϕ ( a )) wherein ϕ ( x ) is a Σ -formula in the signature of L and ψ ( x ) is a Π -formula in the signatureof L that may contain parameters, and x abbreviates ( x , . . . , x n ) , and R is an n -ary conceptvariable for n ≥ that does not appear free in ϕ ( x ) or ψ ( x ) . Finally, traditionally one also includes amongst the predicative systems the following choiceprinciple:
Definition 2.5.
Suppose that L is an expansion of L . The Σ -Choice Schema for L -formulas consists of all axioms of the form (2.4) [ ∀ x ∃ R (cid:48) ϕ ( R (cid:48) , x )] → ∃ R [ ∀ x ϕ ( R [ x ] , x )] wherein the L -formula ϕ ( R (cid:48) , x ) is Σ , perhaps with parameters, and x abbreviates ( x , . . . , x m ) and R is an ( m + n ) -ary concept variable for n, m ≥ that does not appear free in ϕ ( R (cid:48) , x ) where R (cid:48) is an n -ary concept variable. The Σ -Choice Schema and the First-Order Comprehension Schema together imply the ∆ -Comprehension Schema (cf. [26] Theorem V.8.3 pp. 205-206, [29] Proposition 6 p. 1683).Hence, even if one’s primary interest is in the latter schema, typically theories are axiomatizedwith the two former schemas since they are deductively stronger, and that is how we proceedin this paper. 6o the signature L of the weak background second-order logic, we want to associate acertain weak background L -theory. Some of the axioms of this background theory axioma-tize the behavior of the predication symbols and the projection symbols. For each m ≥ extensionality axiom , wherein R, S are m -ary concept variables and a = a , . . . , a m are object variables:(2.5) ∀ R, S [ R = S ↔ ( ∀ a ( R ( a ) ↔ S ( a )))]But it should be noted that some authors don’t explicitly include the identity symbol forconcepts or higher-order entities and simply take it as an abbreviation for coextensionality(cf. [26] pp. 2-3, [3] pp. 14-15). Second, for each n, m ≥
1, one has the following projectionaxioms governing the behavior of the projection symbols, wherein R is an ( m + n )-ary conceptvariable and a = a , . . . , a m , b = b , . . . , b n are object variables:(2.6) ∀ R ∀ a, b [( R [ a ])( b ) ↔ R ( a, b )]Finally, with all this in place, we can define the weak background theory of second-orderlogic: Definition 2.6.
The weak background theory of second-order logic Σ - OS is L -theory con-sisting of (i) the extensionality axioms (2.5) and the projection axioms (2.6) and (ii) the Σ -Choice Schema for L -formulas (Definition 2.5) and (iii) the First-Order ComprehensionSchema for L -formulas (Definition 2.3). In the theory Σ - OS and its extensions, we use standard abbreviations for various operationson concepts, for instance X ∩ Y = { z : Xz & Y z } and { x } = { z : z = x } and X × Y = { ( x, y ) : Xx & Y y } and ∅ = { x : x (cid:54) = x } . In general, we use { x : Φ( x ) } as an abbreviationfor the concept F such that F x iff Φ( x ), assuming that Φ( x ) is a formula which falls underone of the comprehension principles available in the theory in which we are working.This weak background theory Σ - OS of second-order logic is used to define the followingFregean theory at issue in this paper. If E ( R, S ) is an L -formula with two free n E -aryconcept variables and no further free variables, then we let Equiv( E ) abbreviate the L -sentence expressive of E being an equivalence relation on n E -ary concepts, i.e. the universalclosure of the following, wherein R, S, T are n E -ary concept variables:(2.7) [ E ( R, R ) & ( E ( R, S ) → E ( S, R )) & (( E ( R, S ) & E ( S, T )) → E ( R, T ))]Then consider the following collection of L -formulas which consists of all the L -formulas E ( R, S ) with two free n E -ary concept variables and no further free variables such that Σ - OS proves Equiv( E ):(2.8) ProvEquiv( L ) = { E ( R, S ) is an L formula : Σ - OS (cid:96) Equiv( E ) } Then define the following expansion of L of L : Definition 2.7.
Let L consist of the expansion of the signature L (2.1) by a new functionsymbol ∂ E from n E -ary concepts to objects for each E from ProvEquiv( L ) (2.8). Definition 2.8.
The predicative Fregean theory , abbreviated
PFT , is the L -theory consistingof (i) the extensionality axioms (2.5) and the projection axioms (2.6) and (ii) the Σ -ChoiceSchema for L -formulas (Definition 2.5) and (iii) the First-Order Comprehension Schemafor L -formulas (Definition 2.3), and (iv) the abstraction principle A [ E ] (2.1) for each E from ProvEquiv( L ) (2.8). Hence, the theory
PFT is a recursively enumerable theory in a recursively enumerable sig-nature L . If one desired a recursive signature, one could alternatively define L to consistof function symbols ∂ E from n E -ary concepts to objects for each L -formula E , regardlessof whether it was in ProvEquiv( L ) (2.8). This is because clause (iv) in Definition 2.8only includes the abstraction principle A [ E ] (2.1) when the formula E is in fact in the setProvEquiv( L ) (2.8).While this definition is technically precise, the niceties ought not obscure the intuitive-ness of the motivating idea. For, the idea behind this predicative Fregean theory is thatit conjoins traditional predicative constraints on comprehension together with the idea thatabstraction principles associated to certain L -formulae are always available. More capa-ciously: if we start from weak background theory of second-order logic Σ - OS and if we canprove in this theory that an L -formula E ( R, S ) in the signature of this weak backgroundlogic is an equivalence relation on n E -ary concepts for some n E ≥
1, then the predicativeFregean theory
PFT includes the abstraction principle A [ E ] (2.1) associated to E . Hence thetheory PFT includes the abstraction principles associated to number, extension, and ordinal,namely Hume’s Principle (1.1), Basic Law V (1.4) and the abstraction principle associatedto ordinals (cf. (2.2) above).One of the aims of the earlier paper [31] was to establish the following:
Theorem 2.9.
The theory
PFT is consistent.Proof.
Let E , . . . , E n , . . . enumerate the elements of the collection ProvEquiv( L ) from equa-tion (2.8). By compactness, it suffices to establish, for each n ≥
1, the consistency of thesubsystem of
PFT which is formed by restricting part (iv) of the Definition of
PFT to theabstraction principles A [ E ] , . . . , A [ E n ]. But then this theory is a subtheory of the theorywhich, in the paper [31], we called Σ − [ E , . . . , E n ] A + SO + GC . The consistency of this theorywas established in the Joint Consistency Theorem of that paper. While the predicative Fregean Theory only explicitly includes predicative instances of thecomprehension schema for L -formulas, surprisingly it is able to deductively recover all in-stances of the Full Comprehension Schema for L -formulas. Theorem 3.1.
PFT proves each instance of the Full Comprehension Schema for L -formulas. roof. Let Φ( x, G ) be an L -formula with all free variables displayed, wherein x is an objectvariable and G is a unary concept variable. Let us first show that PFT proves the followinginstance of the Full Comprehension Schema for L -formulas (Definition 2.2):(3.1) ∀ G ∃ F ∀ x ( F x ↔ Φ( x, G ))After we finish the proof of this instance, we’ll comment on how to establish the generalcase.First consider the following L -formulas µ ( R, S ) , ν ( R, S ) with all free variables displayed,where
R, S are binary concept variables: µ ( R, S ) ≡ [ ∃ ! x, G with R = { x } × G ] & [ ∃ ! y, H with S = { y } × H ]& ∀ x, G, y, H [( R = { x } × G & S = { y } × H ) → (Φ( x, G ) ↔ Φ( y, H ))] ν ( R, S ) ≡ ¬ [ ∃ ! x, G with R = { x } × G ] & ¬ [ ∃ ! y, H with S = { y } × H ]In this, the identity R = { x } × G is an abbreviation for the claim that(3.2) ∀ a, b ( R ( a, b ) ↔ (( a = x ) & Gb ))Hence, µ ( R, S ) expresses that R can be written uniquely as { x } × G for some x, G , while S can be written uniquely as { y } × H for some y, H , and that Φ( x, G ) ↔ Φ( y, H ). Thecircumstance in which a binary relation R can be written as { x } × G but not uniquely sois when G is empty, since in this case { x } × G = { x (cid:48) } × G for any objects x, x (cid:48) . Finally,consider the following L -formula E ( R, S ) where again
R, S are binary concept variables andall free variables are displayed:(3.3) E ( R, S ) ≡ ( µ ( R, S ) ∨ ν ( R, S ))The weak background theory Σ - OS proves that E ( R, S ) is an equivalence relation onbinary concepts. For reflexivity, either R can be written uniquely as { x } × G for some x, G ,or not. If so, then one trivially has Φ( x, G ) ↔ Φ( x, G ). This then implies µ ( R, R ) and so E ( R, R ). If not, then of course ν ( R, R ) and so E ( R, R ). For symmetry, it simply suffices tonote that both µ and ν are symmetric in that µ ( R, S ) implies µ ( S, R ) and likewise for ν . Fortransitivity, suppose that E ( R, S ) and E ( S, T ). Because of the disjunctive definition of E in (3.3), there are three cases to consider. First suppose that µ ( R, S ) and µ ( S, T ). Then wemay uniquely write R = { x }× G, S = { y }× H, T = { z }× I , and from Φ( x, G ) ↔ Φ( y, H ) andΦ( y, H ) ↔ Φ( z, I ) we may conclude that Φ( x, G ) ↔ Φ( z, I ). Hence we then have µ ( R, T )and thus E ( R, T ). Second suppose that ν ( R, S ) and ν ( S, T ). These two assumptions implythat we can’t write any of
R, S, T uniquely as the product of a singleton and a unary concept,and hence that ν ( R, T ) and E ( R, T ). Finally, suppose that µ ( R, S ) and ν ( S, T ) (or vice-versa). But this case leads to a contradiction, since µ ( R, S ) implies that we can write S uniquely as the product of a singleton and a unary concept, while ν ( S, T ) says that we can’t.Hence E ( R, S ) is indeed an equivalence relation on binary concepts, and provably so in theweak background theory Σ - OS . 9hen the L -formula E ( R, S ) is in the set ProvEquiv( L ) (2.8). Hence the theory PFT contains the abstraction principle A [ E ] (2.1). Before we verify (3.1), let us introduce anotherabstraction principle. Consider the following L -formulas µ (cid:48) ( X, Y ) , ν (cid:48) ( X, Y ) with all freevariables displayed, where
X, Y are unary concept variables: µ (cid:48) ( X, Y ) ≡ ∃ x ∃ y X = { x } & Y = { y } & (Φ( x, ∅ ) ↔ Φ( y, ∅ )) ν (cid:48) ( X, Y ) ≡ ¬ ( ∃ x X = { x } ) & ¬ ( ∃ y Y = { y } )Then consider the following L -formula E (cid:48) ( X, Y ) where again
X, Y are unary concept vari-ables and all free variables are displayed:(3.4) E (cid:48) ( X, Y ) ≡ ( µ (cid:48) ( X, Y ) ∨ ν (cid:48) ( X, Y ))By the same argument as the previous paragraph, Σ - OS proves that E (cid:48) ( X, Y ) is an equiva-lence relation unary concepts. So the theory
PFT contains the abstraction principle A [ E (cid:48) ] (2.1)Now, working in PFT , let us verify (3.1). There are three cases. First suppose that thereis no x with Φ( x , G ). Then to establish (3.1) one can take F = ∅ .As a second case, suppose that there is a x with Φ( x , G ) and that G is non-empty.Then observe that the graph of the function f ( x ) = ∂ E ( { x } × G ) has both a Σ - and aΠ -definition: f ( x ) = y ↔ ∃ R ( ∀ a, b R ( a, b ) ↔ ( a = x & Gb )) & ∂ E ( R ) = y ↔ ∀ R ( ∀ a, b R ( a, b ) ↔ ( a = x & Gb )) → ∂ E ( R ) = y (3.5)These are equivalent because we can use the First-Order Comprehension Schema for L -formulas to secure that the binary relation R = { x }× G exists. Hence by the ∆ -ComprehensionSchema for L -formulas, the equivalence in (3.5) implies that the graph of f exists as a bi-nary concept. Then by First-Order Comprehension Schema for L -formulas, the followingunary concept exists:(3.6) F = { x : f ( x ) = ∂ E ( { x } × G ) } Now let’s argue that F = { x : Φ( x, G ) } . First suppose that F x . Then f ( x ) = ∂ E ( { x } × G )and hence ∂ E ( { x } × G ) = ∂ E ( { x } × G ). Then E ( { x } × G, { x } × G ) and since G is non-empty we have µ ( { x } × G, { x } × G ). Then Φ( x, G ) ↔ Φ( x , G ). Since we’re assumingthat Φ( x , G ), we then conclude that Φ( x, G ), which is what we wanted to show. For theconverse, suppose that Φ( x, G ). Since we’re assuming that Φ( x , G ) and that G is non-empty we may conclude that µ ( { x } × G, { x } × G ) and thus E ( { x } × G, { x } × G ) and ∂ E ( { x } × G ) = ∂ E ( { x } × G ). By the definition of f , we then have f ( x ) = ∂ E ( { x } × G )which by the definition of F implies that F x , which is what we wanted to show.As a third case, suppose that there is an x with Φ( x , G ) but that G itself is empty.Then we argue as before that the graph of g ( x ) = ∂ E (cid:48) ( { x } ) exists as a binary concept, that F = { x : g ( x ) = ∂ E (cid:48) ( { x } ) } exists as a unary concept, and that F = { x : Φ( x, G ) } .This finishes the proof of (3.1) in PFT . The proof of the general case of the Full Compre-hension Schema for L -formulas (Definition 2.2) differs only in that unary concept variable F n -ary concept variable and there may be more than one con-cept parameter G , as well as some additional object parameters. But the proof of this generalcase is directly analogous to the proof of (3.1). The only difference is that the number ofabstraction principles used in the proof will increase with the number of concept parameters.In general if there are m -concept parameters G , . . . , G m , then there will be 2 m different ab-straction principles used in the proof, since one must consider a case corresponding to thefinite binary sequence ( i , . . . , i m ), wherein i k = 0 indicates that G k is empty, and i k = 1indicates that G k is non-empty.Before turning to the proof that PFT interprets second-order Peano arithmetic, let’s brieflynote that in the consistency proof from [31] invoked in the proof of Theorem 2.9, we explicitlyverified the Full Comprehension Schema for L -formulas. (In the language of that paper,these were part of the theory SO , and the interested reader may consult the proof of the JointConsistency Theorem in that paper).While the theory PFT only explicitly includes some instances of the Full ComprehensionSchema for L -formulas in its definition (cf. Definition 2.8), the previous theorem says thatit proves all of them. However, even in this predicative setting, the Russell paradox canbe used to show that there is no concept consisting of the extensions, i.e. the range ofthe extension operator ∂ from Basic Law V (1.4). For a proof, see [29] Proposition 29 p.1692. Now the formula rng( ∂ ) is definable by a Σ -formula of the signature L [ ∂ ]. Further L [ ∂ ] is included in the signature L of PFT . Hence, since the L -theory PFT is consistentby Theorem 2.9, it follows that
PFT does not prove all instances of the Full ComprehensionSchema for L -formulas.This kind of situation is of course not entirely unfamiliar. For instance, Presburgerarithmetic yields a complete axiomatization of the structure ( Z , , , < ) (cf. Marker [24]pp. 82 ff). So this axiomatization proves each instance of the following induction schema inthe signature L = { , , + , < } :(3.7) [ ϕ (0) & ∀ x ≥ ϕ ( x ) → ϕ ( x + 1)))] → [ ∀ x ≥ ϕ (0)]Consider a non-standard model G = ( G, , , + , < ) of Presburger arithmetic, and extend L to L (cid:48) by adding a new unary predicate Z which is interpreted on G as the integers Z . Thenof course the axioms of Presburger arithmetic do not imply all instances of the schema (3.7)in the expanded signature L (cid:48) . So of course it’s consistent for there to be a schema and an L (cid:48) -theory and a subsignature L of L (cid:48) such that the theory proves all instances of the L -schemabut not every instance of the L (cid:48) -schema.Now let’s show that PFT interprets second-order Peano arithmetic PA . These axioms arethe natural set of axioms used to describe the standard model of second-order arithmetic;see [26] p. 4 or [29] p. 1680 or [30] p. 106 for an explicit list of these axioms. Theorem 3.2.
The predicative Fregean theory
PFT interprets second-order Peano arithmetic PA .Proof. First note that the predicative Fregean theory
PFT proves the existence of the graphof the function s ( x ) = ∂ ( { x } ) (cf. [29] Proposition 27 p. 1691), where this is the abstraction11perator associated to Basic Law V (1.4). For, note that in PFT , for all objects x, y , one hasthat the following Σ -condition and Π -conditions are equivalent:(3.8) [ ∃ X ( X = { x } & ∂ ( X ) = y )] ↔ [ ∀ X ( X = { x } → ∂X = y )]By the ∆ -Comprehension Schema for L -formulas, there is then a binary relation whichholds of objects x, y iff either the Σ -condition holds or the Π -condition holds. And thisbinary relation is obviously the graph of the function s ( x ) = ∂ ( { x } ).Let M be { x : x = x } , which exists by Full Comprehension for L -formulas, and let0 = ∂ ( ∅ ). Then one has that the triple ( M, , s ) satisfies the first two axioms of Robinson’s Q :(3.9) ∀ x s ( x ) (cid:54) = 0 , ∀ x, y ( s ( x ) = s ( y ) → x = y )For, suppose that s ( x ) = 0. Then ∂ ( { x } ) = ∂ ( ∅ ) and then by Basic Law V (1.4) one hasthat { x } = ∅ , a contradiction. Similarly, suppose that s ( x ) = s ( y ). Then ∂ ( { x } ) = ∂ ( { y } )and so by Basic Law V (1.4) one has that { x } = { y } and hence x = y . Thus (3.9) followsimmediately from Basic Law V (1.4).But then standard arguments allow one to interpret second-order Peano arithmetic PA bytaking the natural numbers N to be the sub-concept of M consisting of all those subconceptsof M which are “inductive,” that is which contain zero and closed under successor. Hereof course for the existence of N and the verification of the other axioms of arithmetic, oneappeals to the Full Comprehension Schema for L -formulas, using M, , s as parameters (cf.[29] Theorem 16 p. 1688). However, in spite of its technical strength, the conceptual basis of the predicative Fregeantheory
PFT is rather fragile. For, the L -theory PFT was formed by adding the abstractionprinciple A [ E ] associated to the L -formulas E ( R, S ) when this formula could be proven tobe an equivalence relation in the background second-order logic Σ - OS . But one cannot suc-cessively iterate this idea. For, suppose that in analogue to ProvEquiv( L ) in equation (2.8),one defines:(4.1) ProvEquiv( L ) = { E ( R, S ) is an L formula : PFT (cid:96)
Equiv( E ) } And further suppose that one defines L to be the expansion of L by the addition ofa function symbol ∂ E from n E -ary concepts to objects for each L -formula E ( R, S ) inProvEquiv( L ). Finally, suppose one defines the following iteration of PFT (cf. Defini-tion 2.8):
Definition 4.1.
The theory
PFT is the L -theory consisting of (i) the extensionality ax-ioms (2.5) and the projection axioms (2.6) and (ii) the Σ -Choice Schema for L -formulas (Def-inition 2.5) and (iii) the First-Order Comprehension Schema for L -formulas (Definition 2.3), nd (iv) the abstraction principle A [ E ] (2.1) for each E which is from ProvEquiv( L ) (2.8)or from ProvEquiv( L ) (4.1). Then the same argument as in the proof of Theorem 3.1 establishes that
PFT proves eachinstance of the Full Comprehension Schema for L -formulas. But then PFT is inconsistent,since on pain of the Russell paradox there is no concept of all extensions (cf. [29] Proposition29 p. 1692), where again the extensions are the range of the abstraction operator ∂ associatedto Basic Law V (1.4). Hence, while the predicative Fregean theory PFT is consistent, whenone tries to iterate its underlying idea of adding abstraction principles when their equivalencerelations can be proven to be equivalence relations, one again runs up against the Russellparadox. This indicates that the resource of abstraction principles in the predicative settingis unlike that of typed theories of truth or second-order logic, which we may consistently addto any consistent theory.This point is underscored when one observes that the same considerations show theinconsistency of an axiom-based analogue of the rule-based predicative Fregean theory
PFT .In particular, suppose that we recursively defined a signature L ∗ extending L so that if E ( R, S ) is an L ∗ -formula in exactly two free n E -ary concept variables then L ∗ also containsa function symbol ∂ E which takes n E -ary concepts to objects and which does not occur in E . One could then define the following L ∗ -theory: Definition 4.2.
The theory
PFT ∗ is the L ∗ -theory consisting of (i) the extensionality ax-ioms (2.5) and the projection axioms (2.6) and (ii) the Σ -Choice Schema for L ∗ -formulas (Def-inition 2.5) and (iii) the First-Order Comprehension Schema for L ∗ -formulas (Definition 2.3),and (iv) the axiom Equiv( E ) → A [ E ] for each L ∗ -formula E . In this, Equiv( E ) is the sentence which says that E is an equivalence relation (cf. (2.7)) and A [ E ] is the abstraction principle (2.1), so that the axiom Equiv( E ) → A [ E ] says that if E is an equivalence relation, then A [ E ] holds. The considerations of the previous paragraphscan be replicated in this theory PFT ∗ , showing it to be inconsistent. However, the conceptualdistance between the inconsistent L ∗ -theory PFT ∗ and the consistent L -theory PFT is ratherslim. The difference is merely a difference between a rule and an axiom: whereas the rule-based
PFT only includes an abstraction principle when the underlying equivalence relation isexpressible in the weak background logic and is provably an equivalence relation there, theaxiom-based
PFT ∗ includes a commitment to either the truth of the abstraction principle orthe falsity of its underlying formula being an equivalence relation.In response to this, one might try to restrain the predicative Fregean theory PFT so thatthe analogously defined iterated version of it and the analogously defined axiom-based versionof it were consistent. For instance, one might consider restricting the abstraction principlesadded to the theory
PFT to those whose underlying equivalence relation was expressible bothas a Σ -formula and a Π -formula in the background second-order logic. This, it might besuggested, would be a genuinely predicative theory of abstraction principles. Such a movewould block the proof of Theorem 3.1. For, the equivalence relation E ( R, S ) (3.3) used inthat proof is not obviously expressible in such a way. However, it is unknown to us howmuch arithmetic this more austerely predicative theory could interpret, and it is not obvious13o us whether the analogously defined iterated version of it (or axiom-based version of it) isconsistent.Another way forward might be to find some principled way to focus attention on abstrac-tion principles which are somehow more like the paradigmatic Basic Law V (1.4) and Hume’sPrinciple (1.1) and the abstraction principle associated to ordinals (2.2), and somehow lesslike the seemingly ad-hoc abstraction principles constructed in the proof of Theorem 3.1.But to do so would be to lose some of the original motivation for focusing on predicativeabstraction principles. For, part of the attraction was supposed to be that more abstractionprinciples became consistent and jointly consistent. And indeed, as the predicative FregeanTheory
PFT attests, a good deal of joint consistency is available in this setting. Hence in theearlier paper [31] we said that we had resolved an analogue of the joint consistency problem.But as we have seen in this section, when we try to iterate the underlying idea of abstrac-tion principles in the predicative setting, we again run into inconsistency and seem back inthe situation of trying to discern ways to weed out the acceptable from the unacceptableabstraction principles. For an overview of the various candidates for acceptable abstractionprinciples in the general impredicative setting, see [23] or [5].Perhaps another way forward might be to give up on the idea of abstraction principlesaltogether and find principled reasons for studying systems centered around either BasicLaw V (1.4) itself or Hume’s Principle (1.1) itself or the abstraction principle associated toordinals (2.2) all by itself. With respect to Basic Law V (1.4), this is the perspective of[32], where the idea is to work within an intensional logic and see the extension operator asselecting a sense for each concept, just like we might select a specific Turing machine indexfor each computable function. But much remains unknown about the individual abstractionprinciples at the predicative level. For instance, it is to our knowledge unknown whetherBasic Law V (1.4) or the abstraction principle associated to ordinals (2.2), equipped withthe Σ -choice schema and the First-Order Comprehension Schema, interprets the analogouspredicative versions of arithmetic (cf. [29] p. 1707). In this paper, the idea for interpretingarithmetic was to collect together all the predicative abstraction principles so that they couldeffect the interpretation together, and it is in general unclear to us what happens when onefocuses on the abstraction principles one by one. Acknowledgements
I was lucky enough to be able to present parts of this work at a number of workshops andconferences, and I would like to thank the participants and organizers of these events forthese opportunities. I would like to especially thank the following people for the commentsand feedback: Robert Black, Roy Cook, Matthew Davidson, Walter Dean, Marie Duˇz´ı,Kenny Easwaran, Fernando Ferreira, Martin Fischer, Rohan French, Salvatore Florio, Ken-taro Fujimoto, Jeremy Heis, Joel David Hamkins, Volker Halbach, Ole Thomassen Hjortland,Luca Incurvati, Daniel Isaacson, J¨onne Kriener, Graham Leach-Krouse, Hannes Leitgeb,Øystein Linnebo, Paolo Mancosu, Richard Mendelsohn, Tony Martin, Yiannis Moschovakis,John Mumma, Pavel Pudl´ak, Sam Roberts, Marcus Rossberg, Tony Roy, Gil Sagi, Flo-rian Steinberger, Iulian Toader, Gabriel Uzquiano, Albert Visser, Kai Wehmeier, Philip Welch,14revor Wilson, and Martin Zeman.Finally, a special debt is owed to the editors and anonymous referees of this journal, towhom I express my gratitude. For, the proofs were greatly simplified by their suggestionsand the previous reliance upon choice was removed by virtue of these suggestions. Whilecomposing this paper, I was supported by a Kurt G¨odel Society Research Prize Fellowshipand by Øystein Linnebo’s European Research Council funded project “Plurals, Predicates,and Paradox.”
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