scholarly journals NOTES ON BOUNDED INDUCTION FOR THE COMPOSITIONAL TRUTH PREDICATE

2017 ◽  
Vol 10 (3) ◽  
pp. 455-480 ◽  
Author(s):  
BARTOSZ WCISŁO ◽  
MATEUSZ ŁEŁYK
Keyword(s):  
Reflection Principle ◽  
Peano Arithmetic ◽  
Truth Predicate ◽  
Base Theory ◽  
Modified Theory ◽  
Induction Scheme

AbstractWe prove that the theory of the extensional compositional truth predicate for the language of arithmetic with Δ0-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.

scholarly journals Maximal sets and fragments of Peano arithmetic

1989 ◽  
Vol 115 ◽  
pp. 165-183 ◽  
Author(s):  
C.T. Chong
Keyword(s):  
Recursion Theory ◽  
Peano Arithmetic ◽  
Base Theory ◽  
Abstract Principle ◽  
Priority Method ◽  
General Program ◽  
Fragments Of Peano Arithmetic

This work is inspired by the recent paper of Mytilinaios and Slaman [9] on the infinite injury priority method. It may be considered to fall within the general program of the study of reverse recursion theory: What axioms of Peano arithmetic are required or sufficient to prove theorems in recursion theory? Previous contributions to this program, especially with respect to the finite and infinite injury priority methods, can be found in the works of Groszek and Mytilinaios [4], Groszek and Slaman [5], Mytilinaios [8], Slaman and Woodin [10]. Results of [4] and [9], for example, together pinpoint the existence of an incomplete, nonlow r.e. degree to be provable only within some fragment of Peano arithmetic at least as strong as P- + IΣ2. Indeed an abstract principle on infinite strategies, such as that on the construction of an incomplete high r.e. degree, was introduced in [4] and shown to be equivalent to Σ2 induction over the base theory P- + IΣ0 of Peano arithmetic.

Iterated reflection principles and the ω-rule

1982 ◽  
Vol 47 (4) ◽  
pp. 721-733 ◽  
Author(s):  
Ulf R. Schmerl
Keyword(s):  
Reflection Principle ◽  
Peano Arithmetic ◽  
Equivalent System ◽  
Formal Similarity ◽  
Sound System ◽  
Formal Systems ◽  
Ordinal Notations ◽  
Axiomatizable Theory ◽  
Reflection Principles ◽  
Image Position

The ω-rule,with the meaning “if the formula A(n) is provable for all n, then the formula ∀xA(x) is provable”, has a certain formal similarity with a uniform reflection principle saying “if A(n) is provable for all n, then ∀xA(x) is true”. There are indeed some hints in the literature that uniform reflection has sometimes been understood as a “formalized ω-rule” (cf. for example S. Feferman [1], G. Kreisel [3], G. H. Müller [7]). This similarity has even another aspect: replacing the induction rule or scheme in Peano arithmetic PA by the ω-rule leads to a complete and sound system PA∞, where each true arithmetical statement is provable. In [2] Feferman showed that an equivalent system can be obtained by erecting on PA a transfinite progression of formal systems PAα based on iterations of the uniform reflection principle according to the following scheme:Then T = (∪dЄ, PAd, being Kleene's system of ordinal notations, is equivalent to PA∞. Of course, T cannot be an axiomatizable theory.

On maximal theories

1991 ◽  
Vol 56 (3) ◽  
pp. 885-890 ◽  
Author(s):  
Zofia Adamowicz
Keyword(s):  
Open Problem ◽  
Initial Segment ◽  
Peano Arithmetic ◽  
Finite Collection ◽  
Extension Problem ◽  
Recursive Theory ◽  
Image Position ◽  
Induction Scheme

Let S be a recursive theory. Let a theory T* consisting of Σ1 sentences be called maximal (with respect to S) if T* is maximal consistent with S, i.e. there is no Σ1 sentence consistent with T* + S which is not in T*.A maximal theory with respect to IΔ0 was considered by Wilkie and Paris in [WP] in connection with the end-extension problem.Let us recall that IΔ0 is the fragment of Peano arithmetic consisting of the finite collection of algebraic axioms PA− together with the induction scheme restricted to bounded formulas.The main open problem concerning the end-extendability of models of IΔ0 is the following:(*) Does every model of IΔ0 + BΣ1 have a proper end-extension to a model of IΔ0?Here BΣ1 is the following collection scheme:where φ runs over bounded formulas and may contain parameters.It is well known(see [KP]) that if I is a proper initial segment of a model M of IΔ0, then I satisfies IΔ0 + BΣ1.For a wide discussion of the problem (*) see [WP]. Wilkie and Paris construct in [WP] a model M of IΔ0 + BΣ1 which has no proper end-extension to a model of IΔ0 under the assumption IΔ0 ⊢¬Δ0H (see [WP] for an explanation of this assumption). Their model M is a model of a maximal theory T* where S = IΔ0.Moreover, T*, which is the set Σ1(M) of all Σ1 sentences true in M, is not codable in M.

On Diophantine equations solvable in models of open induction

1990 ◽  
Vol 55 (2) ◽  
pp. 779-786 ◽  
Author(s):  
Margarita Otero
Keyword(s):  
Diophantine Equations ◽  
Main Lemma ◽  
Peano Arithmetic ◽  
The Real ◽  
Real Closure ◽  
Open Induction ◽  
Induction Scheme

AbstractWe consider IOpen, the subsystem of PA (Peano Arithmetic) with the induction scheme restricted to quantifier-free formulas.We prove that each model of IOpen can be embedded in a model where the equation has a solution. The main lemma states that there is no polynomial f{x,y) with coefficients in a (nonstandard) DOR M such that ∣f(x,y) ∣ < 1 for every (x,y) Є C, where C is the curve defined on the real closure of M by C: x2 + y2 = a and a > 0 is a nonstandard element of M.

MODELS OF PT– WITH INTERNAL INDUCTION FOR TOTAL FORMULAE

2016 ◽  
Vol 10 (1) ◽  
pp. 187-202 ◽  
Author(s):  
CEZARY CIEŚLIŃSKI ◽  
MATEUSZ ŁEŁYK ◽  
BARTOSZ WCISŁO
Keyword(s):  
Peano Arithmetic ◽  
Truth Predicate ◽  
Saturated Model ◽  
Internal Induction ◽  
Philosophical Project ◽  
Positive Truth ◽  
Recursively Saturated ◽  
Recursively Saturated Model

AbstractWe show that a typed compositional theory of positive truth with internal induction for total formulae (denoted by PTtot) is not semantically conservative over Peano arithmetic. In addition, we observe that the class of models of PA expandable to models of PTtot contains every recursively saturated model of arithmetic. Our results point to a gap in the philosophical project of describing the use of the truth predicate in model-theoretic contexts.

Disquotational truth and analyticity

2001 ◽  
Vol 66 (4) ◽  
pp. 1959-1973 ◽  
Author(s):  
Volker Halbach
Keyword(s):  
Special Kind ◽  
Reflection Principle ◽  
Truth Predicate

Abstract.The uniform reflection principle for the theory of uniform T-sentences is added to PA. The resulting system is justified on the basis of a disquotationalist theory of truth where the provability predicate is conceived as a special kind of analyticity. The system is equivalent to the system ACA of arithmetical comprehension. If the truth predicate is also allowed to occur in the sentences that are inserted in the T-sentences. yet not in the scope of negation, the system with the reflection schema for these T-sentences assumes the strength of the Kripke-Feferman theory KF. and thus of ramified analysis up to ε0.

A reflection principle and its applications to nonstandard models

1995 ◽  
Vol 60 (4) ◽  
pp. 1137-1152
Author(s):  
James H. Schmerl
Keyword(s):  
Number Theory ◽  
Reflection Principle ◽  
Peano Arithmetic ◽  
Second Order ◽  
Order Number ◽  
Nonstandard Models ◽  
Consistent Extension ◽  
Weak Theories

Some methods of constructing nonstandard models work only for particular theories, such as ZFC, or CA + AC (which is second order number theory with the choice scheme). The examples of this which motivated the results of this paper occur in the main theorems of [5], which state that if T is any consistent extension of either ZFC0 (which is ZFC but with only countable replacement) or CA + AC and if κ and λ are suitably chosen cardinals, then T has a model which is κ-saturated and has the λ-Bolzano-Weierstrass property. (Compare with Theorem 3.5.) Another example is a result from [12] which states that if T is any consistent extension of CA + AC and cf (λ) > ℵ0, then T has a natural λ-Archimedean model. (Compare with Theorem 3.1 and the comments following it.) Still another example is a result in [6] in which it is shown that if a model of Peano arithmetic is expandable to a model of ZF or of CA, then so is any cofinal extension of . (Compare with Theorem 3.10.) Related types of constructions can also be found in [10] and [11].A reflection principle will be proved here, allowing these constructions to be extended to models of many other theories, among which are some exceedingly weak theories and also all of their completions.

scholarly journals TRUTH AND FEASIBLE REDUCIBILITY

2019 ◽  
Vol 85 (1) ◽  
pp. 367-421
Author(s):  
ALI ENAYAT ◽  
MATEUSZ ŁEŁYK ◽  
BARTOSZ WCISŁO
Keyword(s):  
Polynomial Time ◽  
Computable Function ◽  
Peano Arithmetic ◽  
Base Theory ◽  
Speed Up ◽  
Image Position

AbstractLet ${\cal T}$ 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 ${\cal T}$ is PA (Peano arithmetic). We establish the following theorem, which implies that ${\cal T}$ has no more than polynomial speed-up over PA.Theorem.${\cal T}$is feasibly reducible to PA, in the sense that there is a polynomial time computable function f such that for every${\cal T}$-proof π of an arithmetical sentence ϕ, f (π) is a PA-proof of ϕ.

scholarly journals The interpretability logic of Peano arithmetic

1990 ◽  
Vol 55 (3) ◽  
pp. 1059-1089 ◽  
Author(s):  
Alessandro Berarducci
Keyword(s):  
Modal Logic ◽  
Modal Analysis ◽  
Set Theory ◽  
Decision Procedure ◽  
Peano Arithmetic ◽  
Relevant Modal Logic ◽  
Binary Operator ◽  
Base Theory ◽  
Interpretability Logic ◽  
Work Done

AbstractPA is Peano arithmetic. The formula InterpPA(α, β) is a formalization of the assertion that the theory PA + α interprets the theory PA + β (the variables α and β are intended to range over codes of sentences of PA). We extend Solovay's modal analysis of the formalized provability predicate of PA, PrPA(x), to the case of the formalized interpretability relation InterpPA(x, y). The relevant modal logic, in addition to the usual provability operator ‘□’, has a binary operator ‘⊳’ to be interpreted as the formalized interpretability relation. We give an axiomatization and a decision procedure for the class of those modal formulas that express valid interpretability principles (for every assignment of the atomic modal formulas to sentences of PA). Our results continue to hold if we replace the base theory PA with Zermelo-Fraenkel set theory, but not with Gödel-Bernays set theory. This sensitivity to the base theory shows that the language is quite expressive. Our proof uses in an essential way earlier work done by A. Visser, D. de Jongh, and F. Veltman on this problem.

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