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Predicativity, the Russell-Myhill Paradox, and Church’s Intensional Logic

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Abstract

This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church’s intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also model other axioms of Church’s intensional logic that have been criticized by Parsons and Klement: this, it turns out, is due to resources which also permit an interpretation of a fragment of Gallin’s intensional logic. Finally, the relation between the predicative response to the Russell-Myhill paradox of propositions and the Russell paradox of sets is discussed, and it is shown that the predicative conception of set induced by this predicative intensional logic allows one to respond to the Wehmeier problem of many non-extensions.

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Notes

  1. The Typed Sense Determines Reference (2.4) is satisfied because the definition of Δ a (f ) in Eq. 2.9 is clearly functional since f is by stipulation a function defined on worlds and the world of evaluation w 0 is fixed. For the Typed Composition Axiom (2.5), suppose that Δ a b (f )=f and Δ a x =x. Then one can calculate that Δ b (f x 〉)=(f x 〉)(w 0)=(f (w 0))(x (w 0))=(Δ a b f )(Δ a x )=f(x).

  2. Admittedly, there is something deeper going on here. The distinction between Alternative (0) and Alternative (1) lies in whether lambda-conversion preserves sense. However, lambda-terms are an alternative way of formalizing comprehension (cf. (4.1)) which the predicative response offered here does not have available in full generality. Thus the formal extensions of Church’s core system with which we work here simply don’t have lambda-terms in the object-language. Hence an immediate issue which faced Church– namely whether to say that lambda-conversion preserves sense– is not even available in the object-language of our systems.

  3. More specifically see Section 500 p. 538 of Russell [78] and p. 82 of Myhill [63]. According to the history as set out in de Rouilhan [19], Russell never mentioned this paradox again. As for Myhill, in the same 1958 paper he reports that Carnap’s “general approach to the problem, in terms of ‘possible worlds’ and state-descriptions, is in [his] opinion practically certain to yield a correct explication within a few years” ([63] p. 81). This contrasts with Myhill’s earlier 1952 paper on Church ([62]) in which he weighs carefully the costs and benefits of Fregean and modal approaches without indicating a decisive preference for either. It is well-known that Myhill continued to work on intuitionistic and non-classical approaches to the set-theoretic paradoxes throughout his career, but to my knowledge he never after the 1958 paper returned to this proposition-theoretic paradox.

  4. Intensional logics like Church’s intensional logic and possible worlds semantics have resources for axiomatizing the notion of a “proposition denoting the true.” In Church’s system, this is written as Δ t (p)=1 while in possible worlds semantics this is written p(w 0)=1 where w 0 is the world of evaluation. However, in neither of these intensional logics does one have the resources for going from a name of a sentence to the proposition expressed by the sentence. If one did, then since these systems of intensional logic are consistent with the addition of resources needed to effect self-reference, one could replicate the formal versions of the liar paradox.

  5. Klement suggests that this kind of concern is one way of understanding Frege’s own reservations about the Russell-Myhill paradox ([52] p. 183).

  6. The Surjectivity Axiom has a long and complicated history in Church’s own writings. In 1946, Church seemed to indicate that Cantor-like paradoxes would lead one to deny this axiom ([14] p. 31). In 1974, Church indicated that this axiom followed from the premises of his system called Alternative 2 ([16] p. 145). In his last paper in 1993, Church included this axiom in his system ([18] pp. 144–145), albeit without saying anything explicit about his reasons for this inclusion. For other statements of the Surjectivity Axiom in the secondary literature, see Anderson [1] principle (C) p. 221 and Klement [52] Theorem LSD(0) 1 p. 116 and Klement [53] p. 305 Axiom PC. For more on Anderson and Klement on the Surjectivity Axiom, see the discussion below.

  7. It is admittedly somewhat inaccurate to speak of definite descriptions merely as an “interpretation of Fregean sense,” since they in fact provide a systematic way of dispensing with the Fregean notion of expression altogether and maintaining that reference is the sole semantic primitive. But presumably part of our tradition’s reason for thinking that Frege’s theory of meaning is susceptible to modal counterexamples couched in terms of definite descriptions is something like the thought that we can think of Fregean senses as definite descriptions.

  8. Cf. [27] pp. 96, 102, 179 ff, [46] pp. 66 ff, [88] p. 323. This idea is also associated to Tichý. See in particular the papers “Sense and Procedure” and “Intensions in Terms of Turing Machines” in [91].

  9. Obviously, one way to respond to the version of the Russell-Myhill paradox formalized here would be to deny the Senses are Objects Axiom (3.9). One way to do that might be to accept that senses are definite descriptions or procedures but to deny that these can be identified with specific objects like Gödel numbers of formulas or indexes of Turing machines. Traditional reasons for such a denial might be that e.g. abstract procedures aren’t represented by a specific index for a specific Turing machine, but rather by a large class of such indexes (cf. [8]). I don’t think that such a response would ultimately succeed. For, grant all this and then just select, for each abstract procedure, a specific index for a specific Turing machine which represents it, and call these things quasi-senses. Then quasi-senses are objects and so one could run the entire Russell-Myhill paradox again with respect to quasi-senses. For, the other axioms occurring in the formalized version of the paradox seem just as plausible for the so-defined quasi-senses as for senses qua abstract procedures. A similar point can be made with respect to definite descriptions simply by selecting Gödel numbers of specific formulas.

  10. See axiom “PCE” on [53] p. 305. Another way of formalizing the system of Klement [53] might be to regard it simply as an untyped system, where there is no distinction between concepts and objects.

  11. It also bears mentioning that Church, Anderson, and Klement additionally considered formalizations of the Russell-Myhill paradox within an alternative framework of intensional logic that goes under the heading of “Russellian intensional logic” ([3, 17, 52] pp. 175 ff). This is the general framework which Klement employed in his widely-read [55]. Since this framework was designed to be an alternative to what we’re calling “Church’s intensional logic,” we have not made use of this in our formalization. By the same token, it is beyond the scope of this paper to say whether anything like a predicative response is available in this alternative framework. To say anything definitive would require at least another lengthy consistency proof like that we offer in Section 5. If it turned out that nothing like a predicative response was available in this alternative setting, this might well indicate a certain lack of robustness to the predicative response offered in this paper.

  12. More formally, we suppose that the types are assigned to sets as follows, wherein E and W are fixed sets, corresponding to the objects and the worlds respectively (cf. (5.1)):

    $$ D_{e} = E, \quad D_{t}=\{0,1\}, \quad D_{ab}=D_{b}^{D_{a}} = \{f: D_{a}\rightarrow D_{b}\}, \quad D_{a^{\prime}} = {D_{a}^{W}} = \{f: W\rightarrow D_{a}\} $$
    (3.13)

    Suppose that we are working in a set-theoretic metatheory where as usual |X| is used to denote the cardinality of the set X. Then either \(\left |D_{t^{\prime }}\right |< \left |D_{e}\right |\) or not. If so, then there is no injection from D e to \(D_{t^{\prime }}\) and the Propositions as Fine-Grained as Objects Axiom (3.10) comes out false. Suppose alternatively that \(\left |D_{t^{\prime }}\right |\geq \left |D_{e}\right |\). Since we’re working in a set-theoretic metatheory, we can then appeal to Cantor’s theorem and basic facts about cardinality to obtain that \(\left | D_{(t^{\prime }t)^{\prime }}\right | \geq \left |D_{t^{\prime }t}\right | >\left |D_{t^{\prime }}\right | \geq \left |D_{e}\right |\). Then it is not the case that \(D_{(t^{\prime }t)^{\prime }}\) (or anything bijective with it) is a subset of D e and hence the Senses are Objects Axiom (3.9) comes out false for the specific type of a=(t t).

  13. Obviously other approaches have been neglected as well: for instance, I know of no extant approaches to the Russell-Myhill paradox which adopt the perspective of non-classical logic.

  14. A distinct set of motivations for predicativity constraints come from the apparent affinity of predicativity with types of constructivism. For more on this complicated aspect of the history of predicativity, see Parsons [66]. Another important study of the history of predicativity-like conceptions is Goldfarb’s [42] study of Russell. Goldfarb suggests that Russell’s reasons for endorsing predicativity-like constraints might be related to having systems in which one can quantify over intensional entities like propositions. By contrast, the motivations given here for predicativity constraints are not intended to have anything to do with constructivity and are intended to apply with equal force to the quantifiers ranging over intensional entities like propositions as to those ranging over extensional entities like concepts.

  15. Presumably this motivation for the restriction on the quantifiers in the Predicative Typed Comprehension Schema (4.6) likewise motivates the restriction on the higher-order parameters in this schema. In this it’s helpful to recall the worked-out example above of the higher-order parameters \(\mathcal {D}\) and \(\mathcal {C}\). As one can see by inspecting Eqs. 4.10 and 4.11 above, higher-order parameters are able to go proxy for higher-order quantifiers. Given this, if one wants to employ a description featuring a higher-order parameter to stably refer to a lower-order entity, then it’s natural to require that this higher-order parameter likewise not shift in extension under variations of the range of the higher-order quantifiers.

  16. The language of “positions” is apt because, as one can see from inspection of the below definitions, one intensional position can provide the interpretation of the n-th order quantifiers in one intensional hierarchy but the interpretation of the m-th order quantifiers in another.

  17. It’s worth spelling out exactly how one defines \(\mathcal {O}_{n}\) and π n , by more specific reference to the details of the Existence Theorem of [94] and in particular to the function 𝜃 n defined therein. The simplest way is to take \(\mathcal {O}_{n}=\theta _{n}^{-1}(L_{\alpha _{n}})\) and to define \(\pi _{n}=\theta _{n}\upharpoonright \mathcal {O}_{n}\). Since \(\theta _{n}:\mathcal {F}_{n}\dashrightarrow L_{\alpha _{n}}\) is \(\underset {\sim }{\Sigma }{\,\!}_{n}^{L_{\alpha _{n}}}\)-definable and \(\mathcal {F}_{n}\) is \(\underset {\sim }{\Sigma }{\,\!}_{1}^{L_{\alpha _{n}}}\)-definable, \(\mathcal {O}_{n}\) will be \(\underset {\sim }{\Sigma }{\,\!}_{n}^{L_{\alpha _{n}}}\)-definable, and the total surjective map \(\pi _{n}: \mathcal {O}_{n}\rightarrow L_{\alpha _{n}}\) will be similarly definable. Because it is total, trivially \(\mathcal {O}_{n}\setminus \pi _{n}^{-1}(L_{\alpha _{n}})\) is \(\underset {\sim }{\Sigma }{\,\!}_{n}^{L_{\alpha _{n}}}\)-definable because it is, well, empty. Since π n is designed to provide the interpretation of Δ a for each type a of degree n (cf. subsequent discussion circa Eq. 5.13), clearly this interpretation clashes with the intended interpretation of the presentation functions, on which they would be partial. To reinstitute partiality, choose any subset \(\mathcal {P}_{n}\subseteq \mathcal {F}_{n}\setminus \theta ^{-1}_{n}(L_{\alpha _{n}})\) which is \(\underset {\sim }{\Sigma }{\,\!}_{n}^{L_{\alpha _{n}}}\)-definable and then define \(\mathcal {O}^{\prime }_{n}=\theta _{n}^{-1}(L_{\alpha _{n}}) \cup \mathcal {P}_{n}\) and define \(\pi _{n}^{\prime }=\theta _{n}\upharpoonright \mathcal {O}^{\prime }_{n}\), making sure to build the parameters defining \(\mathcal {P}_{n}\) into ν n . For instance, one could take \(\mathcal {P}_{n}\) to be any finite subset of \(\mathcal {F}_{n}\setminus \theta ^{-1}_{n}(L_{\alpha _{n}})\).

  18. To see this, suppose that the antecedent of the Iterative Axiom (6.1) held, so that Δ a b (f ) was defined and not equal to Δ a b (g ). Then let f a b (f ) and let g a b (g ). Since f,g are functional entities of type ab, it must be the case that they differ on some value (cf. (2.6)), so that there is an object x of type a such that f(x)≠g(x). Now consider the representation x =∇ a (x) of this entity x. Then by the Representation Axiom (6.4), we have that x presents x, or that Δ a (x )=x. Now, using the Characterization of Intensional Application (6.6), let us quickly compute f x 〉 and g x 〉:

    $$\begin{array}{@{}rcl@{}} f^{\prime}\langle x^{\prime}\rangle=\nabla_{b}(({\Delta}_{ab}(f^{\prime})({\Delta}_{a}(x^{\prime}))) = \nabla_{b} (f(x)) \end{array} $$
    (6.10)
    $$\begin{array}{@{}rcl@{}} g^{\prime}\langle x^{\prime}\rangle=\nabla_{b}(({\Delta}_{ab}(g^{\prime})({\Delta}_{a}(x^{\prime}))) = \nabla_{b} (g(x)) \end{array} $$
    (6.11)

    Now, to finish the verification of the Iterative Axiom (6.1), suppose for the sake of contradiction that Δ b (f x 〉)=Δ b (g x 〉). Then by the previous calculations, we see that Δ b (∇ b (f(x)))=Δ b (∇ b (g(x))). By the Representation Axiom (6.4), it then follows that f(x)=g(x), contrary to hypothesis. Hence, this is why the axioms pertaining to the representation function deductively imply the Iterative Axiom (6.1).

  19. First suppose that f is injective and that f presents f and that we have the identity f x 〉=f y 〉. By the Characterization of Intensional Application (6.6), we then have the identity

    $$ \nabla_{b}(({\Delta}_{ab}(f^{\prime}))({\Delta}_{a}(x^{\prime}))) = \nabla_{b}(({\Delta}_{ab}(f^{\prime}))({\Delta}_{a}(y^{\prime}))) $$
    (6.13)

    But since the representation function ∇ b is an injection and since f presents f, this reduces to the identity f a (x ))=f a (y )) and since f is an injection, we have Δ a (x )=Δ a (y ), which is what we wanted to show. This completes the verification of the left-to-right direction of Eq. 6.12.

    For the right-to-left direction of Eq. 6.12, suppose that f satisfies the right-hand side of Eq. 6.12. Suppose for the sake of contradiction that f is not an injection, so that f(x)=f(y) but xy. Then let x =∇ a (x) and y =∇ a (y) and f =∇ a b (f), so that the Representation Axiom (6.4) implies that x presents x and y presents y and f presents f. Then we can expand the identity f(x)=f(y) to

    $$ ({\Delta}_{ab} f^{\prime})({\Delta}_{a}(x^{\prime})) = f(x) = f(y) = ({\Delta}_{ab} f^{\prime})({\Delta}_{a}(y^{\prime})) $$
    (6.14)

    Then by applying the representation function ∇ b to each side and appealing to the Characterization of the Intensional Application (6.6), we have that f x 〉=f y 〉. Then by the hypothesis that f satisfies the right-hand side of Eq. 6.12, we have that Δ a (x )=Δ a (y ), and since x presents x and y presents y, we have that x=y, which contradicts the reductio assumption that xy. This completes the argument that the Characterization of Intensional Injectivity (6.12) follows from our axioms governing the representation function.

  20. See in particular Klement [54] pp. 165-166. But this is only one aspect of the concern of Parsons and Klement. First, Parsons was most interested in the interaction of Church’s other axiom (6.2) with senses which do not present any referent ([68] p. 517). Second, Klement was also concerned with unintuitive consequences of Eq. 6.2 related to the intentionality of senses (cf. [54] p. 164). A deeper question raised by the work of Parsons and Klement is what analogues there are of the Typed Comprehension Schema (4.1) for type (a b). This is relevant because Parsons and Klement’s counterexamples to Church’s other axiom (6.2) are engendered by combining senses or intensions of type (a b) together in various ways, a procedure which would be most naturally warranted by a version of the comprehension schema for senses or intensions of type (a b).

  21. For instance, some of the systems of Church and Anderson explicitly included axioms for the injectivity of senses of functional expressions. See the axiom designated “64” in Church [16] p. 151 and Anderson [1] p. 222.

  22. However, it should be emphasized that the predicative response, as we have described it above in Section 4, is not necessarily committed to the existence of a sense-selecting extension operator. For, the two proofs from the above paragraphs used the representation operator ∇ a from Section 6 and the Predicative Typed Choice Schema (4.8). As stressed in Section 4, the philosophical motivations for the Predicative Typed Comprehension Schema (4.7) don’t necessarily extend to the Predicative Typed Choice Schema (4.8); and it goes without saying that while the representation operator helps bring more of the model construction into the object-language, it too is not necessarily built into the predicative response to the Russell-Myhill paradox. Indeed, it is not even clear to me whether one can derive the existence of sense-selecting extension operators merely from the core of Church’s system (2.8), the Surjectivity Axiom (3.8), the Senses are Objects Axiom (3.9), the Propositions as Fine-Grained as Objects Axiom (3.10), and the Predicative Typed Comprehension Schema (4.7). Thus the results of this section are only available to certain natural expansions of the predicative perspective by choice principles or by a representation operator.

  23. That said, there are some differences. First, this formalization provides no insight into how, if at all, \(\widetilde {h}\) is provided “by reference” to h. Second, on our explication of “definite”, it will follow that the definite concepts are closed under boolean operations such as intersection, union, and complement. If one has the intuition that “definite concepts” should be small in some sense, one will resist the claim that definite concepts are closed under complementation. Finally, it should be noted that this general variety of formalization of indefinite extensibility is of course not new: see for instance Shapiro-Wright [83] p. 266 and Priest [71] pp. 1264-1265.

  24. The first argument for Eq. 7.9 employs the representation function. So suppose that the sense-selecting extension operator satisfies (f)=∇ e t f as in Eq. 7.2. Fix a parameter q of type et. Then one has the following, wherein x has type e and f has type et:

    $$ \forall \; x \; \exists \; ! \; f \; [(h(x)=0 \; \& \; f=q) \vee (h(x)=1 \; \& \; \nabla_{et}(f)=x)] $$
    (7.10)

    Then by the Predicative Typed Comprehension Schema (4.6), there is γ h of type e(e t) such that

    $$ \forall \; x \; [(h(x)=0 \; \& \; \gamma_{h}(x)=q) \vee (h(x)=1 \; \& \; \nabla_{et}(\gamma_{h}(x))=x)] $$
    (7.11)

    To verify Eq. 7.9, suppose that h((g))=1. Letting x=(g) we have that h(x)=1. Then ∇ e t (γ h (x))=x=(g)=∇ e t g. Then γ h (x)=g and so γ h ((g))=g, which is what we wanted to show. The second argument for Eq. 7.9 employs the Predicative Typed Choice Schema (4.8). Since h⊆rng(), we have that ∀ x [h(x)=1→(∃ f (f)=x)]. Then the definition of sense-selecting (7.5) implies that ∀ x [h(x)=1→(∃ ff Δ e t (f )=ff =x)]. Trivially we then have ∀ xf [h(x)=1→f =x]. Then we may apply the Predicative Typed Choice Schema (4.8) since the parameter h has type with degree 2 and the type e(e t) has degree 2. Doing this we get an entity β h of type e(e t) such that ∀ x [h(x)=1→β h (x)=x]. Further, we claim that

    $$ \forall \; x \; \exists \; f \; [h(x)=1\rightarrow {\Delta}_{et}(\beta_{h}(x))=f] $$
    (7.12)

    For, if h(x)=1 then (f)=x for some f of type et and hence Δ e t (g )=f and g =x for some g of type (e t) by the definition of sense-selecting (7.5). Then β h (x)=x=g and so Δ e t (β h (x))=Δ e t (g )=f. So indeed equation (7.12) holds. Further, we may apply the Predicative Typed Choice Schema (4.1) to this equation since the parameters h,β h have types with degree 2 and since e(e t) likewise has degree 2. Then we obtain γ h of type e(e t) such that

    $$ \forall \; x \; [h(x)=1\rightarrow {\Delta}_{et}(\beta_{h}(x))=\gamma_{h}(x)] $$
    (7.13)

    Let’s now verify Eq. 7.9. Suppose that g is of type et such that h((g))=1. Let x=(g), so that h(x)=1. By (g)=x, we obtain Δ e t (g )=g and g =x for some g of type (e t) by the definition of sense-selecting (7.5). Further by Eq. 7.13, we have Δ e t (β h (x))=γ h (x). Then β h (x)=x=g and so g e t (g )=Δ e t (β h (x))=γ h (x)=γ h ((g)), which is what we wanted to show.

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Acknowledgments

I was lucky enough to be able to present parts of this work at a number of workshops and conferences, and I would like to thank the participants and organizers of these events for these opportunities. I would like to especially thank the following people for the comments and feedback which I received on these and other occasions: Robert Black, Roy Cook, Matthew Davidson, Walter Dean, Marie Duží, Kenny Easwaran, Fernando Ferreira, Martin Fischer, Rohan French, Salvatore Florio, Kentaro Fujimoto, Jeremy Heis, Joel David Hamkins, Volker Halbach, Ole Thomassen Hjortland, Luca Incurvati, Daniel Isaacson, Jönne Kriener, Graham Leach-Krouse, Hannes Leitgeb, Øystein Linnebo, Paolo Mancosu, Richard Mendelsohn, Tony Martin, Yiannis Moschovakis, John Mumma, Pavel Pudlák, Sam Roberts, Marcus Rossberg, Tony Roy, Gil Sagi, Florian Steinberger, Iulian Toader, Gabriel Uzquiano, Albert Visser, Kai Wehmeier, Philip Welch, Trevor Wilson, and Martin Zeman. This paper has likewise been substantially bettered by the feedback and comments of the editors and referees of this journal, to whom I express my gratitude. While composing this paper, I was supported by a Kurt Gödel Society Research Prize Fellowship and by Øystein Linnebo’s European Research Council funded project “Plurals, Predicates, and Paradox.”

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Appendices

Appendix 1: Proof of Proposition on Location of Domains

In this brief appendix, we prove the Proposition on Location of Domains (5.11) from Section 5. The proof is by simultaneous induction on n≥1. For n=1, note that (I) holds vacuously. As for (II), first note that if a is a type with ∥a∥=1, then a is among the types e,e ,e ,…,t,t ,t ,…. Now, if a=e or a=t, then part (II) follows trivially since α 0<α 1 and so both α 0 and the set {0,1} are members of \(L_{\alpha _{1}}\) and μ 1 includes the parameter α 0 by definition. Suppose the result holds for a. Since \(\mathcal {O}_{1}\) is a \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{1}}\)-definable subset of \(L_{\alpha _{1}}\) in parameter ν 1, it follows trivially that \(D_{a^{\prime }}=\mathcal {O}_{1}\) is \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{1}}\)-definable subset of \(L_{\alpha _{1}}\) in the more complex parameter μ 1. This completes the argument in the case n=1.

Now suppose that the result holds for n, and we show it holds for n+1. For (I), suppose that a is a type with ∥a∥<n+1, say ∥a∥=m. Then since D a is a definable subset of \(L_{\alpha _{m}}\) by the induction hypothesis on part (II) for m, we may write \(D_{a} = \{x\in L_{\alpha _{m}}: L_{\alpha _{m}}\models \psi (x,\mu _{m})\}\) for some formula ψ. Hence D a is an element of \(L_{\alpha _{m}+1}\) and a member of \(L_{\alpha _{n+1}}\). Then we have that D a is the unique X in \(L_{\alpha _{n+1}}\) which satisfies the following condition:

$$ (\forall \; x\in X\cap L_{\alpha_{m}} \; L_{\alpha_{m}}\models \psi(x,\mu_{m})) \; \& \; (\forall \; x\in L_{\alpha_{m}} \; (L_{\alpha_{m}}\models \psi(x,\mu_{m})\rightarrow x\in X)) $$
(9.1)

Then since m<n+1 and the parameter μ n+1 contains the parameter μ m as well as the ordinal α m , and since the map βL β is Δ1 in \(L_{\alpha _{n+1}}\) (cf. [51] II.2.8 p. 70) and since the satisfaction relation is likewise Δ1 (cf. [51] I.9.10 p. 41), this is a Σ1-condition in \(L_{\alpha _{n+1}}\) in parameter μ n+1. Here we’re also appealing tacitly to the fact that the Σ1-formulas are closed under bounded quantification in models L α which satisfy Σ1-collection (cf. [51] Lemma I.11.6 p. 53). This completes the induction step for part (I) of of the proposition.

For the induction step for part (II), note that the types with degree n+1 are of the form a or ab. Then we may do a subinduction on complexity of type. First suppose that ∥a ∥=n+1 and suppose that the result holds for a; we show it holds for a . Since \(\mathcal {O}_{n+1}\) is \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}\)-definable subset of \(L_{\alpha _{n+1}}\) in parameter ν n+1, it follows trivially that \(D_{a^{\prime }}=\mathcal {O}_{n+1}\) is a \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}\)-definable subset of \(L_{\alpha _{n+1}}\) in the more complex parameter μ n+1.

Second suppose that ∥a b∥=n+1, and suppose that the result holds for a,b; we show it holds for ab. There are two subcases here. In the first subcase, suppose that ∥a∥≥∥b∥. Then by the definition of degree in Eq. 4, we have that ∥a∥,∥b∥<∥a b∥. Then if we let fnct(f) abbreviate the Σ0-formula expressive of the graph f being functional, and fixing similar Σ0-definitions of dom(f)=X and rng(f)⊆Y, then the set \(({D_{b}}^{D_{a}}) \cap L_{\alpha _{n+1}}\) is equal to

$$ \{f\!\in\! L_{\alpha_{n+1}}\!: \text{fnct}(f) \; \& \; \exists \; X,Y \; X=D_{a} \; \& \; \text{dom}(f)=X \; \& \; Y = D_{b} \; \& \; \text{rng}(f)\subseteq Y \} $$
(9.2)

Then by part (I), we have that this is a \(\underset {\sim }{\Sigma }{\,\!}_{1}\)-definable subset of \(L_{\alpha _{n+1}}\) in parameter μ n+1.

Now, as a second subcase, suppose the result holds for a,b and that ∥a∥<∥b∥, so that by the definition of degree in Eq. 4 we have n+1=∥a b∥=∥b∥. Then D a is a member of \(L_{\alpha _{n+1}}\) by part (I), while by the supposition that the result holds for b we have that D b is a \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}\)-definable subset of \(L_{\alpha _{n+1}}\) in parameter μ n+1. Then \(({D_{b}}^{D_{a}}) \cap L_{\alpha _{n+1}}\) is also a \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}\)-definable subset of \(L_{\alpha _{n+1}}\) in the parameter μ n+1. For, we have the following definition of \(({D_{b}}^{D_{a}}) \cap L_{\alpha _{n+1}}\):

$$ \{f\in L_{\alpha_{n+1}}: L_{\alpha_{n+1}}\models [\text{fnct}(f) \; \& \; \exists \; X \; X=D_{a} \; \& \; \forall \; x\in X \; \exists \; y\in D_{b} \; \langle x,y\rangle\in f]\} $$
(9.3)

Here we are appealing to part (I) applied to D a since ∥a∥<n+1 in this subcase. Likewise, we are appealing to the fact that the bounded quantification in the last conjunct does not move us out of the complexity class \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}\) in models of \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}\)-collection and \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}-1}\)-separation. This finishes the induction step for (II). With this the inductive proof of the proposition is finished.

Appendix 2: Verification of the Satisfaction of Predicative Comprehension

Here we prove the Theorem on Consistency of Predicative Comprehension from Section 5 (cf. Eq. 5.18). Further, we here prove this result for the language expanded by the representation functions \(\nabla _{a}: D_{a}\rightarrow D_{a^{\prime }}\) introduced in Section 6 (cf. circa (6.4)). As a first step towards approaching the proof of this theorem, let’s first note an elementary result on terms. The terms in the signature of an intensional structures consists simply of the closure of the constants 0,1 and the variables under the extensional application symbols and the representation operations. The presentation symbols and the intensional application symbols are not total and hence are formally treated as relation symbols as opposed to function symbols. The type of a term is defined inductively as follows: the truth-values 0, 1 have type t, the variables have the type that they are given initially, and if τ has type ab and σ has type a, then τ(σ) or e-app a b (τ,σ) has type b; and if τ has type a then ∇ a (τ) has type a . Then we have the following elementary result:

$$\begin{array}{@{}rcl@{}} &&\text{(Proposition that Terms do not Raise Degree). Suppose that}\,\tau(x_{1}, \ldots, x_{k})\, \text{is a}\\ &&\text{term in the signature of intensional structures with all free variables displayed}\\ &&\text{such that the type of each variable}\,x_{i}\, \text{has degree}\,\leq~n.\, \text{Then the type of the term}\\ &&\tau\, \text{has degree}\,\leq~n. \end{array} $$
(10.1)

The proof is by induction on the complexity of the term. Clearly this is true in the case of the truth-values and the variables. Suppose it holds for τ(x 1,…,x k ) and σ(x 1,…,x k ); we must show it is the case for e-app a b (τ,σ) and ∇ a (τ). First consider the case of e-app a b (τ,σ). Then τ has type ab and σ has type a, and each has type with degree ≤n by the induction hypothesis. There are two cases to consider, corresponding to the two clauses in the definition of ∥a b∥ in Eq. 4. First suppose that ∥a∥≥∥b∥. Then one has that ∥b∥≤∥a∥≤n, which is what we wanted to show since the type of e-app a b (τ,σ) is b. Second suppose that ∥a∥<∥b∥. Then we have that ∥b∥=∥a b∥≤n, which is again what we wanted to show. Finally, consider the case of ∇ a (τ). Then τ has type a, and it has degree ≤n by induction hypothesis. Then ∇ a (τ) has type a and so ∥a ∥=∥a∥≤n by the definition of degree of ∥a ∥ in Eq. 4. This is why terms do not raise degree, or why Eq. 10.1 holds.

Relatedly, as a preliminary step, let’s establish the following result about the complexity of the functions on intensional structures induced by terms:

$$\begin{array}{@{}rcl@{}} &&\text{(Proposition on Complexity of Terms) Suppose that}\,\tau(\overline{u})\equiv \tau(u_{1}, \ldots, u_{j})\, \text{is a}\\ &&\text{term with all free variables displayed where}\,u_{i}\, \text{has type}\,d_{i}.\, \text{Since terms don't raise}\\ &&\text{degree~(10.1),}\,\tau\, \text{has type with degree}\,d\, \text{with}\,\|d\|\leq m=\max\{\|d_{1}\|, \ldots, \|d_{j}\|\}.\\ &&\text{Then}\,\tau\, \text{induces a function}\,\tau^{\mathbb{D}}: D_{d_{1}}\times {\cdots} \times D_{d_{j}}\rightarrow D_{d}\, \text{whose graph is}\\ &&\underset{\sim}{\Sigma}{\,\!}^{L_{\alpha_{m}}}_{\ell_{m}}\text{-definable}. \end{array} $$
(10.2)

Clearly this is the case if the term is variable. Now for the induction step suppose that the result holds for τ and σ; we must show it holds for \(\rho (\overline {u})\equiv \text {e-app}_{e_{1}e_{2}}(\tau (\overline {u}), \sigma (\overline {u}))\). Then \(\tau (\overline {u})\) has type e 1 e 2 and \(\sigma (\overline {u})\) has type e 1. Since terms don’t raise degree (10.1), it follows that ∥e 1 e 2∥,∥e 1∥ are all less than or equal to m= max{∥d 1∥,…,∥d j ∥}, and from this we infer that ∥e 2∥≤∥e 1 e 2∥≤m as well. Then \(\rho ^{\mathbb {D}}: D_{d_{1}}\times {\cdots } \times D_{d_{j}}\rightarrow D_{e_{2}}\) has the following graph:

$$\begin{array}{@{}rcl@{}} {}\{ (\overline{u},u)\in D_{d_{1}}\times {\cdots} \times D_{d_{j}}\times D_{e_{2}} & \!\!\!:\!\! & \exists \; y\in D_{e_{1}} \; \exists \; z \in D_{e_{1}e_{2}} \; \\ & & \!\!\!\!\sigma^{\mathbb{D}}(\overline{u})= y\; \!\&\! \; \tau^{\mathbb{D}}(\overline{u})=z \; \& \; \langle y,u\rangle \in z\} \end{array} $$
(10.3)

This is \(\underset {\sim }{\Sigma }{\,\!}^{L_{\alpha _{m}}}_{\ell _{m}}\)-definable by the Location of Domains (5.11) since we have that ∥e 1∥,∥e 2∥,∥e 1 e 2∥≤m. For the final induction step, suppose that the result holds for τ; we must show it holds for ∇ a (τ). Then τ has type a, and since terms don’t raise degree (10.1), it follows that ∥a∥≤m. Then the graph of \(\tau ^{\mathbb {D}}\) is \(\underset {\sim }{\Sigma }{\,\!}^{L_{\alpha _{m}}}_{m}\)-definable by induction hypothesis. Recall from the discussion in Section 6 that the representation function ∇ a is interpreted on intensional structures \(\mathbb {D}\) by the function ι a from the definition of an intensional hierarchy (5.9). However, this was by definition \(\underset {\sim }{\Sigma }{\,\!}^{L_{\alpha _{\|a\|}}}_{\|a\|}\)-definable (cf. clause (iii) of the definition of an intensional position (5.8)). Since ∥a∥≤m, we then have that the composition \(\iota _{\|a\|}\circ \tau ^{\mathbb {D}}\) is clearly also \(\underset {\sim }{\Sigma }{\,\!}^{L_{\alpha _{m}}}_{m}\)-definable. This finishes the proof of result on the complexity of terms (10.2).

Now let’s consider what kinds of symbols can appear in a formula covered by the Predicative Typed Choice Schema (4.8). Suppose that the formula φ(x,y,z 1,…,z k ) is a formula with all free variables displayed and with free variable x of type a, y of type b, and in addition variable z i has type c i with ∥c i ∥≤∥a b∥ and all the bound variables in φ(x,y,z 1,…,z k ) have type c with ∥c∥<∥a b∥. Let ∥a b∥=n+1. There are then two cases to consider, corresponding to the split in cases in the definition of the degree ∥a b∥ in (4). If ∥a∥≥∥b∥, then n+1=∥a b∥=∥a∥+1 and so ∥b∥≤∥a∥≤n. Further, if we split the parameter variables z 1,…,z k into those that have type with degree n+1 and those that have type with degree ≤ n, then we can write the formula in question as:

$$\begin{array}{@{}rcl@{}} &&\text{(First Configuration):}\,\varphi(x, y, v_{1}, \ldots, v_{m}, z_{1}, \ldots, z_{k})\, \text{is a formula with all free}\\ &&\text{variables displayed and with free variable}\,x\, \text{of type}\,a\, \text{with}\,\|a\|\leq{n},\,y\, \text{of type}\,b\\ &&\text{with}\,\|b\|\leq{n},\, \text{and in addition variable}\,v_{i}\, \text{has type}\,a_{i}\, \text{with}\,\|a_{i}\|\leq{n}\, \text{and variable}\\ &&z_{i}\, \text{has type}\,c_{i}\, \text{with}\,\|c_{i}\|=n+1\, \text{and all the bound variables in the formula have}\\ &&\text{type}\,c\, \text{with}\,\|c\|\leq{n}. \end{array} $$
(10.4)

Alternatively, in the other case, we have ∥a∥<∥b∥ and n+1=∥a b∥=∥b∥. If we again split the parameter variables z 1,…,z k into those that have type with degree n+1 and those that have type with degree ≤ n, then we can write the formula in question as:

$$\begin{array}{@{}rcl@{}} &&\text{(Second Configuration):}\,\varphi(x, v_{1}, \ldots, v_{m}, y, z_{1}, \ldots, z_{k})\, \text{is a formula with all free}\\ &&\text{variables displayed and with free variable}\,x\, \text{of type}\,a\, \text{with}\,\|a\|\leq{n},\,y\, \text{of type}\,b\\ &&\text{with}\,\|b\|= n+1,\, \text{and in addition variable}\,v_{i}\, \text{has type}\,a_{i}\, \text{with}\,\|a_{i}\|\leq{n}\, \text{and}\\ &&\text{variable}\,z_{i}\, \text{has type}\,c_{i}\, \text{with}\,\|c_{i}\|=n+1\, \text{and all the bound variables in the formula}\\ &&\text{have type}\,c\, \text{with}\,\|c\|\leq{n}. \end{array} $$
(10.5)

For ease of future reference, we call these two kinds of formulas which can feature in the Predicative Typed Choice Schema (4.8) the “first configuration” and the “second configuration”.

The plan in what follows is to show that the Predicative Typed Choice Schema (4.8) holds for formulas in the second configuration (10.5), and then to show it for formulas in the first configuration (10.4). This first step is done by proving a result connecting the satisfaction of a formula in the second configuration to a certain level of definability in the constructible hierarchy. To build up to the statement of this result, suppose that φ(x,v 1,…,v m ,y,z 1,…,z k ) is in the second configuration (10.5). Then any subformula of this formula has the form

$$ \psi(x,v_{1}, \ldots, v_{m}, v_{m+1}, {\ldots} v_{m+m^{\prime}},y,z_{1}, \ldots, z_{k}) $$
(10.6)

where the variable v i for i>m has type a i with degree ≤n. Let’s abbreviate \(\overline {v} = \langle v_{1}, \ldots , v_{m}, v_{m+1}, {\ldots } v_{m+m^{\prime }}\rangle \) and let’s abbreviate

$$ D_{\overline{a}}=D_{a_{1}}\times {\cdots} \times D_{a_{m+m^{\prime}}}, \hspace{10mm} D_{\overline{c}} = D_{c_{1}}\times {\cdots} \times D_{c_{k}} $$
(10.7)

Note that since ∥a∥,∥a i ∥≤n, it follows from the Location of Domains (5.11), we have that \(D_{a}\times D_{\overline {a}}\) is a member of \(L_{\alpha _{n+1}}\). However, since ∥b∥,∥c i ∥=n+1, we have that \(D_{b}\times D_{\overline {c}}\) is a \(\underset {\sim }{\Sigma }{\,\!}^{L_{\alpha _{n+1}}}_{\ell _{n+1}}\)-definable subset of \(L_{\alpha _{n+1}}\). Having put this terminology in place, let’s now show that:

$$\begin{array}{@{}rcl@{}} &&\text{(Proposition on Complexity of Satisfaction, Second Configuration) For every}\\ &&\text{intensional hierarchy}\,D\, \text{with induced intensional structure}\,\mathbb{D}\, \text{and every subformula}\\ &&\psi(x,\overline{v},y,\overline{z})\, \text{of a formula in the second configuration (10.5), the following set is}\\ &&\underset{\sim}{\Sigma}{\,\!}_{\ell_{n+1}}^{L_{\alpha_{n+1}}}\text{-definable:} \end{array} $$
(10.8)
$$[ \psi]^{D}= \{(x,\overline{v},y,\overline{z})\in D_{a}\times D_{\overline{a}}\times D_{b}\times D_{\overline{c}} : \mathbb{D}\models \psi(x,\overline{v},y,\overline{z})\} $$

We establish this by induction on the complexity of the subformula. By pushing all the negations to the inside, it suffices to show that the result holds for atomics, negated atomics, and is closed under conjunctions, disjunctions, existential quantification, and universal quantification. Let’s begin with the atomic case, considering the negated atomic cases along the way. The atomic formulas in intensional structures have three possible forms, namely:

$$ \tau = \sigma, \hspace{10mm} {\Delta}_{d}(\tau)=\sigma, \hspace{10mm} \text{i-app}_{c_{0}d_{0}}(\tau, \sigma) = \rho $$
(10.9)

where τ,σ,ρ are terms. These are the only possible subformulas because technically, the second is shorthand for the binary atomic formula Δ d (τ,σ) and the third is shorthand for the associated ternary atomic relation (cf. discussion circa Eqs. 5.13 and 5.15). Since τ,σ,ρ appear in a formula in the second configuration (10.5), the free variables in these terms τ,σ,ρ have types with degree ≤n+1 and since terms don’t raise degree (10.1), it follows that the respective types e 1,e 2,e 3 of τ,σ,ρ are also such that ∥e 1∥,∥e 2∥,∥e 3∥≤n+1. From this it follows in turn that ∥d∥≤n+1 and ∥c 0∥,∥d 0∥≤∥c 0 d 0∥=∥(c 0 d 0)∥≤n+1.

Let’s consider first the case of equality between terms, that is, atomic formulas of the form

$$ \psi(x,\overline{v}, y,\overline{z})\equiv \tau(x,\overline{v},y,\overline{z})=\sigma(x,\overline{v},y, \overline{z}) $$
(10.10)

Then we have that

$$\begin{array}{@{}rcl@{}} (x,\overline{v},y,\overline{z})\in [\psi]^{D} & \Longleftrightarrow & \exists \; z_{1}\in D_{e_{1}}, \exists \; z_{2}\in D_{e_{2}} \; \tau(x,\overline{v},y,\overline{z})=z_{1} \\ && \; \& \; \sigma(x,\overline{v},y,\overline{z})=z_{2} \; \& \; z_{1}=z_{2} \end{array} $$
(10.11)

which is \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}^{L_{\alpha _{n+1}}}\)-definable by the result on the complexity of terms (10.2). Similarly we have that

$$\begin{array}{@{}rcl@{}} (x,\overline{v},y, \overline{z})\in [\neg \psi]^{D} & \Longleftrightarrow & \exists \; z_{1}\in D_{e_{1}}, \exists \; z_{2}\in D_{e_{2}} \; \tau(x,\overline{v},y, \overline{z})=z_{1} \\ & & \; \& \; \sigma(x,\overline{v},y, \overline{z})=z_{2} \; \& \; z_{1}\neq z_{2} \end{array} $$
(10.12)

which is \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}^{L_{\alpha _{n+1}}}\)-definable for the same reasons.

Now let’s consider the case of the presentation symbols, that is atomic formulas of the form

$$ \psi(x,\overline{v},y, \overline{z})\equiv {\Delta}_{d}(\tau(x,\overline{v},y,\overline{z}))=\sigma(x,\overline{v},y,\overline{z}) $$
(10.13)

Then [ψ]D is \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}^{L_{\alpha _{n+1}}}\)-definable because we have the following biconditional and because ∥d∥≤n+1 implies that π d is \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}^{L_{\alpha _{n+1}}}\)-definable:

$$\begin{array}{@{}rcl@{}} (x,\overline{v},y,\overline{z})\in [\psi]^{D} & \Longleftrightarrow & \exists \; z_{1}\in D_{e_{1}}, \exists \; z_{2}\in D_{e_{2}} \; \tau(x,\overline{v},y,\overline{z})=z_{1} \\ & & \; \& \; \sigma(x,\overline{v},y,\overline{z})=z_{2} \; \& \; \pi_{\|d\|}(z_{1})=z_{2} \end{array} $$
(10.14)

Further, by using the fact that \(\mathcal {O}_{j}\setminus \pi ^{-1}_{j}(L_{\alpha _{j}})\) was \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{j}}^{L_{\alpha _{j}}}\)-definable for all j≥1 (cf. clause (v) of the definition of an intensional position (5.8)) and that ∥d∥≤n+1, we have that

$$\begin{array}{@{}rcl@{}} (x,\overline{v},y,\overline{z})\in [\neg\psi]^{D} & \Longleftrightarrow& \exists \; z_{1}\in D_{e_{1}}, \exists \; z_{2} \in D_{e_{2}}, \; \exists \; z_{3}\in L_{\alpha_{n+1}} \\ & & \tau(x,\overline{v},y,\overline{z})=z_{1} \; \& \; \sigma(x,\overline{v},y, \overline{z})=z_{2} \\ & & \; \&\! \; (z_{1}\in \mathcal{O}_{\|d\|}\setminus \pi^{-1}_{\|d\|}(L_{\alpha_{\|d\|}})) \vee (\pi_{\|d\|}(z_{1})=z_{3} \; \!\&\! \; z_{2}\neq z_{3}) \end{array} $$
(10.15)

As the final atomic case, consider the case of intensional application:

$$ \psi(x,\overline{v},y, \overline{z})\equiv \text{i-app}_{c_{0}d_{0}}(\tau(x,\overline{v},y,\overline{z}), \sigma(x,\overline{v},y,\overline{z})) = \rho(x,\overline{v},y,\overline{z}) $$
(10.16)

Then the type e 1 of τ must be (c 0 d 0) and the type e 2 of σ must be \(c_{0}^{\prime }\). Then we have

$$\begin{array}{@{}rcl@{}} (x,\overline{v},y,\overline{z})\in [\psi]^{D} & \Longleftrightarrow & \exists \; z_{1}\in D_{(c_{0}d_{0})^{\prime}}, \exists \; z_{2}\in D_{c_{0}^{\prime}}, \exists \; z_{3}\in D_{d_{0}^{\prime}} \\ & & \tau(x,\overline{v},y,\overline{z}) = z_{1} \; \!\&\! \; \sigma(x,\overline{v},y,\overline{z})=z_{2} \; \!\&\! \; \rho(x,\overline{v},y,\overline{z})=z_{3} \\ & & \; \& \; \iota_{\|d_{0}\|}((\pi_{\|c_{0}d_{0}\|}(z_{1}))(\pi_{\|c_{0}\|}(z_{2})) = z_{3} \end{array} $$
(10.17)

Since ∥c 0∥,∥d 0∥≤∥c 0 d 0∥=∥(c 0 d 0)∥≤n+1, we have that \(\pi _{\|c_{0}d_{0}\|}, \pi _{\|c_{0}\|}, \iota _{\|d_{0}\|}\) are all \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}^{L_{\alpha _{n+1}}}\)-definable. Finally, for the negation, we may argue as follows, again appealing to the fact that in the definition of an intensional position we required that the set \(\mathcal {O}_{j}\setminus \pi ^{-1}_{j}(L_{\alpha _{j}})\) was \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{j}}^{L_{\alpha _{j}}}\)-definable for all j≥1 (cf. clause (v) of the definition of an intensional position (5.8)):

$$\begin{array}{@{}rcl@{}} (x,\overline{v},y,\overline{z})\in [\neg \psi]^{D} & \Longleftrightarrow & \exists \; z_{1}\in D_{(c_{0}d_{0})^{\prime}}, \exists \; z_{2} \in D_{c_{0}^{\prime}}, \exists \; z_{3}\in D_{d_{0}^{\prime}}, \exists \; z_{4}\in L_{\alpha_{n+1}} \\ & & \tau(x,\overline{v},y,\overline{z})\! =\! z_{1} \; \!\&\! \; \sigma(x,\overline{v},y,\overline{z})\,=\,z_{2} \; \!\&\! \; \rho(x,\overline{v},y,\overline{z})\,=\,z_{3} \\ & & \wedge [(z_{1}\in \mathcal{O}_{\|c_{0}d_{0}\|}\setminus \pi^{-1}_{\|c_{0}d0\|}(L_{\alpha_{\|c_{0}d_{0}\|}})) \\ &&\; \vee\; (z_{2}\in \mathcal{O}_{\|c_{0}\|}\setminus \pi^{-1}_{\|c_{0}\|}(L_{\alpha_{\|c_{0}\|}})) \\ & & \; \vee \; (\iota_{\|d_{0}\|}((\pi_{\|c_{0}d_{0}\|}(z_{1}))(\pi_{\|c_{0}\|}(z_{2})) = z_{4} \; \& \; z_{4}\neq z_{3})] \end{array} $$
(10.18)

This completes the base cases of the inductive argument for (10.8). Since \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}^{L_{\alpha _{n+1}}}\)-definability is closed under finite intersections and unions, the inductive steps for conjunction and disjunction are trivial. Let us then consider the case of universal quantification. Suppose that the result holds for \(\psi (x,\overline {v},v,y,\overline {z})\) and let us show it holds for \(\theta (x,\overline {v},y,\overline {z})\equiv \forall \; v_{0} \; \psi (x,\overline {v},v_{0},y,\overline {z})\). Since this is a subformula of a formula in the second configuration (10.5), it follows the bound variable v 0 has a type a 0 with degree ≤n. Then by part (ii) of the result on Locations of Domains (5.11), it follows that \(X=D_{a_{0}}\) is a \(\underset {\sim }{\Sigma }{\,\!}^{L_{\alpha _{n+1}}}_{1}\)-condition. Then one has that

$$ (x,\overline{v},y,\overline{z})\in [\theta]^{D} \Longleftrightarrow \exists \; X \; X = D_{a_{0}} \; \& \; \forall \; v_{0}\in X \; (x,v_{0},\overline{v}, y,\overline{z})\in [\psi]^{D} $$
(10.19)

so that [𝜃]D is likewise \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}^{L_{\alpha _{n+1}}}\)-definable since \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}^{L_{\alpha _{n+1}}}\)-definability is closed under bounded quantification in models \(L_{\alpha _{n+1}}\) of \({\Sigma }_{\ell _{n+1}}\)-collection and \({\Sigma }_{\ell _{n+1}-1}\)-separation. A similar argument holds in the case of the existential quantifier, but is even easier since there we do not have to appeal to this result about closure under bounded quantification. This finishes the result on the complexity of satisfaction in the case of a formula which is in the second configuration (10.8).

Now let us finally establish that the Predicative Typed Choice Schema (4.6) holds on intensional structures, at first with respect to formulas in the second configuration (10.5). Suppose that the antecedent holds:

$$ \mathbb{D}\models \forall \;x \; \exists \; y \; \varphi(x,p_{1}, \ldots, p_{m}, y,q_{1}, \ldots, q_{k}) $$
(10.20)

where φ(x,v 1,…,v m ,y,z 1,…z k ) is in the second configuration (10.5). Consider the following relation:

$$ R(x,y)\equiv [x\in D_{a} \; \& \; y\in D_{b} \; \& \; \mathbb{D}\models \varphi(x,p_{1}, \ldots, p_{m},y, q_{1}, \ldots, q_{k})] $$
(10.21)

Then by the result on the complexity of satisfaction (10.8), one has that R is \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}^{L_{\alpha _{n+1}}}\)-definable. And by Eq. 10.20, one has that

$$ L_{\alpha_{n+1}}\models \forall \;x\in D_{a} \; \exists \; y \; R(x,y) $$
(10.22)

By the uniformization theorem (cf. [49] Theorem 3.1 p. 256 and Lemma 2.15 p. 255; [51] Theorem 4.5 p. 269, and “weak uniformization” in [94]), choose a \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}^{L_{\alpha _{n+1}}}\)-definable relation R R such that

$$ L_{\alpha_{n+1}} \models [\forall \; x \; (\exists \; y \; R(x,y))\rightarrow (\exists \; ! \; y \; R^{\prime}(x,y))] $$
(10.23)

Then by Eq. 10.22, one has that R is the graph of a function h:D a D b . Since this graph is \(\underset {\sim }{\Sigma }{\,\!}_{\ell _{n+1}}^{L_{\alpha _{n+1}}}\)-definable with domain D a an element of \(L_{\alpha _{n+1}}\), by Replacement (cf. [51] Lemma I.11.7 p. 53) one has that it is an element of \(L_{\alpha _{n+1}}=L_{\alpha _{\|ab\|}}\). Then h is an element of the domain \(D_{b}^{D_{a}}\cap L_{\alpha _{\|ab\|}}= D_{ab}\) (cf. the third clause of Eq. 5.10). Hence, we’ve shown that there is h in D a b such that

$$ \mathbb{D}\models \forall \;x \; \varphi(x,p_{1}, \ldots, p_{m}, h(x),q_{1}, \ldots, q_{k}) $$
(10.24)

which is what we were required to show in the consequent of the Predicative Typed Choice Schema (4.6).

We’ve verified that the Predicative Typed Choice Schema (4.6) holds on intensional structures, at least with respect to formulas in the second configuration (10.5). Let’s now argue that the same holds with respect to formulas in the first configuration (10.4). Suppose that

$$ \mathbb{D}\models \forall \;x \; \exists \; y \; \varphi(x,y,p_{1}, \ldots, p_{m}, q_{1}, \ldots, q_{k}) $$
(10.25)

where φ(x,y,v 1,…,v m ,z 1,…z k ) is in the first configuration (10.4). Then consider the following, where w is a variable of type ab with degree n+1 and x 1,x 2 are variables of type a:

$$ \psi(x,w,\overline{p}, \overline{q})\!\equiv\! (\forall \; x_{1}, x_{2}\; w(x_{1})\,=\,w(x_{2})) \; \& \; (\exists \; x_{1}, y \; \; (w(x_{1})\,=\,y \; \& \; \varphi(x,y,\overline{p}, \overline{q}))) $$
(10.26)

Intuitively this formula ψ is saying that w is a constant function of type ab and its constant value is a witness to φ. Now ψ is in the second configuration (10.5), and we can verify by hand that for every element y of D b there is a constant function of type ab whose constant value is y. For, if \(y\in L_{\alpha _{\|b\|}}\) then {〈x 1,y〉:x 1D a } is in \(L_{\alpha _{\|ab\|}}\). Then by Predicative Typed Choice Schema (4.6) applied to ψ, we have that there is an element h of type a(a b) such that \(\mathbb {D}\models \forall \; x \; \psi (x,h(x), p_{1}, \ldots , p_{m}, q_{1}, \ldots , q_{k})\). Note that since ∥a b∥=n+1=∥a(a b)∥ we have that h is in \(L_{\alpha _{n+1}}\). Then the function g:D a D b such that g(x)=(h(x))(x) is in \(L_{\alpha _{n+1}}\) by Σ0-separation since

$$ g = \{\langle x,y\rangle\in D_{a}\times D_{b}: \langle x, \langle x,y\rangle\rangle \in h\} $$
(10.27)

We’ve now shown that g is an element of D a b and by construction we have

$$ \mathbb{D}\models \forall \;x \; \varphi(x,g(x),p_{1}, \ldots, p_{m}, q_{1}, \ldots, q_{k}) $$
(10.28)

so that we also have that Predicative Typed Choice Schema (4.6) holds on intensional structures, regardless of which of the two configurations we are in.

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Walsh, S. Predicativity, the Russell-Myhill Paradox, and Church’s Intensional Logic. J Philos Logic 45, 277–326 (2016). https://doi.org/10.1007/s10992-015-9375-5

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