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Local collection scheme and end-extensions of models of compositional truth
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Abstract
We introduce a principle of local collection for compositional truth predicates and show that it is conservative over the classically compositional theory of truth in the arithmetical setting. This axiom states that upon restriction to formulae of any syntactic complexity, the resulting predicate satisfies full collection. In particular, arguments using collection for the truth predicate applied to sentences occurring in any given (code of a) proof do not suffice to show that the conclusion of that proof is true, in stark contrast to the case of induction scheme. We analyse various further results concerning end-extensions of models of compositional truth and the collection scheme for the compositional truth predicate.
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In the following paper we propose a model-theoretical way of comparing the “strength” of various truth theories which are conservative over PA . Let denote the class of models of PA which admit an expansion to a model of theory . We show (combining some well known results and original ideas) that where denotes simply the class of all models of PA and denotes the class of recursively saturated models of PA . Our main original result is that every model of PA which admits an expansion to a model of , admits also an expansion to a model of UTB . Moreover, as a corollary to one of the results (brought to us by Carlo Nicolai) we conclude that UTB is not relatively interpretable in TB , thus answering the question from Fujimoto (Bull Symb Log 16:305–344, 2010).
We prove that the theory of the extensional compositional truth predicate for the language of arithmetic with Δ 0 -induction scheme for the truth predicate and the full arithmetical induction scheme is not conservative over Peano Arithmetic. In addition, we show that a slightly modified theory of truth actually proves the global reflection principle over the base theory.
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.
Given a resplendent model for Peano arithmetic there exists a full satisfaction class over , i.e. an assignment of truth-values, to all closed formulas in the sense of with parameters from , which satisfies the usual semantic rules. The construction is based on the consistency of an appropriate system of -logic which is proved by an analysis of standard approximations of nonstandard formulas.
We present a cut elimination argument that witnesses the conservativity of the compositional axioms for truth (without the extended induction axiom) over any theory interpreting a weak subsystem of arithmetic. In doing so we also fix a critical error in Halbach's original presentation. Our methods show that the admission of these axioms determines a hyper-exponential reduction in the size of derivations of truth-free statements.
This paper develops the model theory of ordered structures that satisfy Keisler's regularity scheme and its strengthening REF script L sign (the reflection scheme) which is an analogue of the reflection principle of Zermelo-Fraenkel set theory. Here script L sign is a language with a distinguished linear order <, and REF script L sign consists of formulas of the form x y1 < x forall yn < x φ y1} dots ,y2)→ φ < x (y1, yn), where φ is an Script L sign -formula, φ <x is the Script L sign -formula obtained by restricting all the quantifiers of φ to the initial segment determined by x, and x is a variable that does not appear in φ. Our results include: The following five conditions are equivalent for a complete first order theory T in a countable language Script L sign with a distinguished linear order: (1) Some model of T has an elementary end extension with a first new element. (2) T REF Script L sign. (3) T has an ω 1-like model that continuously embeds ω 1. (4) For some regular uncountable cardinal κ, T has a κ-like model that continuously embeds a stationary subset of κ. (5) For some regular uncountable cardinal κ, T has a κ-like model M that has an elementary extension in which the supremum of M exists. Moreover, if κ is a regular cardinal satisfying κ = κ <κ , then each of the above conditions is equivalent to: (6) T has a κ + -like model that continuously embeds a stationary subset of κ.
We investigate the notion of definability with respect to a full satisfaction class σ for a model of Peano arithmetic. It is shown that the σ-definable subsets of always include a class which provides a satisfaction definition for standard formulas. Such a class is necessarily proper, therefore there exist recursively saturated models with no full satisfaction classes. Nonstandard extensions of overspill and recursive saturation are utilized in developing a criterion for nonstandard definability. Finally, these techniques yield some information concerning the extendibility of full satisfaction classes from one model to another.
Let be any of the three canonical truth theories CT ⁻ (compositional truth without extra induction), FS ⁻ (Friedman–Sheard truth without extra induction), or KF ⁻ (Kripke–Feferman truth without extra induction), where the base theory of is PA (Peano arithmetic). We establish the following theorem, which implies that has no more than polynomial speed-up over PA.
Theorem. is feasibly reducible to PA, in the sense that there is a polynomial time computable function f such that for every -proof π of an arithmetical sentence ϕ , f ( π ) is a PA -proof of ϕ .
This book analyses and defends the deflationist claim that there is nothing deep about our notion of truth. According to this view, truth is a 'light' and innocent concept, devoid of any essence which could be revealed by scientific inquiry. Cezary Cieśliński considers this claim in light of recent formal results on axiomatic truth theories, which are crucial for understanding and evaluating the philosophical thesis of the innocence of truth. Providing an up-to-date discussion and original perspectives on this central and controversial issue, his book will be important for those with a background in logic who are interested in formal truth theories and in current philosophical debates about the deflationary conception of truth.
It is shown that a nonstandard model of Peano arithmetic which has a full satisfaction class is necessarily recursively saturated.
Preliminaries.- A.- I: Arithmetic as Number Theory, Set Theory and Logic.- II: Fragments and Combinatorics.- B.- III: Self-Reference.- IV: Models of Fragments of Arithmetic.- C.- V: Bounded Arithmetic.- Bibliographical Remarks and Further Reading.- Index of Terms.- Index of Symbols.
By a classical result of Kotlarski, Krajewski and Lachlan, pathological satisfaction classes can be constructed for countable,
recursively saturated models of Peano arithmetic. In this paper we consider the question of whether the pathology can be eliminated;
we ask in effect what generalities involving the notion of truth can be obtained in a deflationary truth theory (a theory
of truth which is conservative over its base). It is shown that the answer depends on the notion of pathology we adopt. It
turns out in particular that a certain natural closure condition imposed on a satisfaction class—namely, closure of truth
under sentential proofs—generates a nonconservative extension of a syntactic base theory (Peano arithmetic).
KeywordsConservativeness-Deflationism-Truth-Satisfaction classes
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