Abstract. We use model-theoretic ideas to present a perspicuous and versatile method of constructing full satisfaction classes on models of Peano arithmetic. We also comment on the ramifications of our work on issues related to conservativity and interpretability.
Abstract. Motivated by Leibniz's thesis on the identity of indiscernibles, Mycielski introduced a set-theoretic axiom, here dubbed the Leibniz-Mycielski axiom LM, which asserts that for each pair of distinct sets x and y there exists an ordinal α exceeding the ranks of x and y, and a formula ϕ(v), such that (V α , ∈) satisfies ϕ(x) ∧ ¬ϕ(y).We examine the relationship between LM and some other axioms of set theory. Our principal results are as follows:1. In the presence of ZF, the following are equivalent:The existence of a parameter free definable class function F such that for all sets x with at least two elements, ∅ = F(x) x. (c) The existence of a parameter free definable injection of the universe into the class of subsets of ordinals.
Abstract. In this paper we examine the relationship between automorphisms of models of I∆ 0 (bounded arithmetic) and strong systems of arithmetic, such as P A, ACA 0 (arithmetical comprehension schema with restricted induction), and Z 2 (second order arithmetic). For example, we establish the following characterization of P A by proving a "reversal" of a theorem of Gaifman: Our results also shed light on the metamathematics of the Quine-Jensen system N F U of set theory with a universal set.
We investigate the structure of fixed point sets of self-embeddings of models of arithmetic. Our principal results are Theorems A, B, and C below.In what follows M is a countable nonstandard model of the fragment IΣ 1 of PA (Peano Arithmetic); N is the initial segment of M consisting of standard numbers of M; I fix (j) is the longest initial segment of fixed points of j; Fix(j) is the fixed point set of j; K 1 (M) consists of Σ 1definable elements of M; and a self-embedding j of M is said to be a proper initial self-embedding if j(M) is a proper initial segment of M.1 Smoryński established the right-to-left direction of this result and left the status of the other, much easier direction as an open problem. It is unclear who first established the easier direction, but by now it is considered part of the folklore of the subject. A different proof of (a stronger version of) Smoryński's theorem was established in [6].2 This result was generalized in [7] by showing that if N is strong in M, then the isomorphism types of fixed point sets of automorphisms of M are precisely the isomorphism types of elementary submodels of M, thus confirming a conjecture of Schmerl.
By a well-known result of Kotlarski, Krajewski, and Lachlan (1981), first-order Peano arithmetic PA can be conservatively extended to the theory CT − [PA] of a truth predicate satisfying compositional axioms, i.e., axioms stating that the truth predicate is correct on atomic formulae and commutes with all the propositional connectives and quantifiers. This results motivates the general question of determining natural axioms concerning the truth predicate that can be added to CT − [PA] while maintaining conservativity over PA. Our main result shows that conservativity fails even for the extension of CT − [PA] obtained by the seemingly weak axiom of disjunctive correctness DC that asserts that the truth predicate commutes with disjunctions of arbitrary finite size. In particular, CT − [PA] + DC implies Con(PA).Our main result states that the theory CT − [PA] + DC coincides with the theory CT 0 [PA] obtained by adding ∆ 0 -induction in the language with the truth predicate. This result strengthens earlier work by Kotlarski (1986) and Cieśliński (2010). For our proof we develop a new general form of Visser's theorem on non-existence of infinite descending chains of truth definitions and prove it by reduction to (Löb's version of) Gödel's second incompleteness theorem, rather than by using the Visser-Yablo paradox, as in Visser's original proof (1989). Fedor Pakhamov's research is supported in part by the Young Russian Mathematics award. 2010 Mathematical Subject Classification: 03F30.CT − [PA] (compositional truth over PA with induction only for the language L A of arithmetic) is conservative over PA, i.e., if an L A -sentence ϕ is provable in CT − [PA], then ϕ is already provable in PA. New proofs of this conservativity result were given by Visser and Enayat [6] using basic model theoretic ideas, and by Leigh [14] using proof theoretic tools; these new proofs make it clear that in the Krajewski-Kotlarski-Lachlan theorem the theory PA can be replaced by any 'base' theory that supports a modicum of coding machinery for handling elementary syntax.On the other hand, it is well-known [9, Thm. 8.39 and Cor. 8.40] that the consistency of PA (and much more) is readily provable in the stronger theory CT [PA], which is the result of strengthening CT − [PA] with the scheme of induction over natural numbers for all L A+T -formulae, where L A+T := L A ∪ {T(x)}. 1 Indeed, it is straightforward to demonstrate the consistency of PA within the subsystem CT 1 [PA] of CT[PA], where CT n [PA] is the subtheory of CT[PA] with the scheme of induction over natural numbers limited to L A+T -formulae that are at most of complexity Σ n [16, Thm. 2.8]. The discussion above leaves open whether CT 0 [PA] is conservative over PA. Kotlarski [11] established that CT 0 [PA] is a subtheory of CT − [PA] + Ref(PA), where Ref(PA) is the L A+T -sentence stating that "every first order consequence of PA is true". Recently Le lyk [15] demonstrated that the converse also holds, which immediately implies that CT 0 [PA] is not conservative over PA since Con(P...
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