Systems of combinatory logic related to Quine's ‘New Foundations’

AbstractSystems TRC and TRCU of illative combinatory logic are introduced and shown to be equivalent in consistency strength and expressive power to Quine's set theory ‘New Foundations’ (NF) and the fragment NFU + Infinity of NF described by Jensen, respectively. Jensen demonstrated the consistency of NFU + Infinity relative to ZFC; the question of the consistency of NF remains open. TRC and TRCU are presented here as classical first-order theories, although they can be presented as equational theories; they are not constructive

On the relative strengths of fragments of collection

Let $\mathbf{M}$ be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, $\Delta_0$-separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set-theoretic collection scheme to $\mathbf{M}$. We focus on two common parameterisations of collection: $\Pi_n$-collection, which is the usual collection scheme restricted to $\Pi_n$-formulae, and strong $\Pi_n$-collection, which is equivalent to $\Pi_n$-collection plus $\Sigma_{n+1}$-separation. The main result of this paper shows that for all $n \geq 1$, (1) $\mathbf{M}+\Pi_{n+1}\textrm{-collection}+\Sigma_{n+2}\textrm{-induction on } \omega$ proves the consistency of Zermelo Set Theory plus $\Pi_{n}$-collection, (2) the theory $\mathbf{M}+\Pi_{n+1}\textrm{-collection}$ is $\Pi_{n+3}$-conservative over the theory $\mathbf{M}+\textrm{strong }\Pi_n \textrm{-collection}$. It is also shown that (2) holds for $n=0$ when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke-Platek Set Theory with Infinity and $V=L$) that does not include the powerset axiom.Comment: 22 page

Automorphisms of models of bounded arithmetic

We establish the following model theoretic characterization of the fragment I∆0 +Exp+BΣ1 of Peano arithmetic in terms of fixed points of automorphisms of models of bounded arithmetic (the fragment I∆0 of Peano arithmetic with induction limited to ∆0-formulae). Theorem A. The following two conditions are equivalent for a countable model M of the language of arithmetic: (a) M satisfies I∆0 + BΣ1 + Exp. (b) M = Ifix(j) for some nontrivial automorphism j of an end extension N of M that satisfies I∆0. Here Ifix(j) is the largest initial segment of the domain of j that is pointwise fixed by j, Exp is the axiom asserting the totality of the exponential function, and BΣ1 is the Σ1-collection scheme consisting of the universal closure of formulae of the form [∀x < a ∃y ϕ(x, y)] → [∃z ∀x < a ∃y < z ϕ(x, y)], where ϕ is a ∆0-formula. Theorem A was inspired by a theorem of Smoryński, but the method of proof of Theorem A is quite different and yields the following strengthening of Smoryński’s result: Theorem B. Suppose M is a countable recursively saturated model of P A and I is a proper initial segment of M that is closed under exponentiation. There is a group embedding j ↦− → ̂j from Aut(Q) int

On the relative strengths of fragments of collection

Let M be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, Δ0‐separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set‐theoretic collection scheme to M. We focus on two common parameterisations of the collection: Πn‐collection, which is the usual collection scheme restricted to Πn‐formulae, and strong Πn‐collection, which is equivalent to Πn‐collection plus Σn+1‐separation. The main result of this paper shows that for all n≥1,M+Πn+1− collection +Σn+2− inductionon ω proves that there exists a transitive model of Zermelo Set Theory plus Πn‐collection,the theory M+Πn+1− collection is Πn+3‐conservative over the theory M+ strong Πn− collection .It is also shown that (2) holds for n=0 when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke‐Platek Set Theory with Infinity plus V=L) that does not include the powerset axiom.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/149299/1/malq201800044.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/149299/2/malq201800044_am.pd