Definable sets and expansions of models of Peano arithmetic

Murawski, Roman

doi:10.1007/bf01625830

Cited by 7 publications

(5 citation statements)

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“…The following theorem is proved in [7]. It was also shown in [7] that if I and J are closed initial segments of M and I c J,…”

Section: Rmurawskjmentioning

confidence: 92%

Some Properties of the Family of Expansions to Models of A/Δ + Σ‐AC

Murawski

1991

Mathematical Logic Qtrly

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In [71 we have studied particular expansions of nonstandard models M of Peano arithmetic PA to models of a fragment T of the second order arithmetic A; which consists of axioms of Peano arithmetic with the axiom of induction, d!-comprehension axiom (A:-CA) and a I;:-axiom scheme of choice (C:-AC) (it is denoted by AJA + Zi-AC). Those expansions consisted of sets definable in a sense of a given nonstandard satisfaction class S over a model M of PA by (possibly nonstandard) formulas from an initial segment I of M with arbitrary parameters from M . We showed some properties of such expansions. In the present paper we shall study other properties of them and their families, in particular we shall consider the problem of isomorphism of expansions of the above type.Let us recall necessary definitions. Throughout the paper A4 denotes a countable nonstandard model of Peano arithmetic PA. We shall use full substitutable satisfaction classes over M (for definitions of those notions cf. e.g. [S], [6], 131). Recall here only that the notion of a satisfaction class over a given model M of PA extends the TARSKI'S definition of satisfaction to the class of nonstandard formulas in the sense of M. A satisfaction class S over M is said to be substitutable iff all substitutions of the axiom scheme of induction by formulas of the language L(PA) u {S} are true in (M, S). A satisfaction class S is said to be full iff it decides all formulas (in the sense of M) on all valuations.Let Fm(.) denote a formula of the language L(PA) of Peano arithmetic which strongly represents in PA the recursive set of Godel numbers of formulas of L(PA). We identify formulas with their Godel numbers (in a given arithmetization of the syntax of L(PA)) and we do not distinguish between logical connectives and quantifiers on the one hand and their counterparts in the arithmetization of the language on the other.Definition. An initial segment I of a model M is said to be closed under logical operations (shortly: closed) iff for any cp, p E I and k E I , if M!= (Frn(cp) & Fm(y)), then ( 7 cp) E I , It can be shown (cf. [8]): (1) If I!=IZl, then I is closed, where 11, denotes the subtheory of PA with the axiom scheme of induction restricted to formulas only; (2) the family of all initial segments Is, M, where Z is closed, is of the order type of the Cantor set 2" with its lexicographical ordering; and hence (3) this family is of the cardinality 2w.Definition. Let I E , M be closed and let S be a satisfaction class over M. We define the family Def,(M,

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“…The following theorem is proved in [7]. It was also shown in [7] that if I and J are closed initial segments of M and I c J,…”

Section: Rmurawskjmentioning

confidence: 92%

Some Properties of the Family of Expansions to Models of A/Δ + Σ‐AC

Murawski

1991

Mathematical Logic Qtrly

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“…On the contrary, Grzegorczyk (2005) suggests "undecidability without arithmetization" if "the decidability is defined directly as the property of graphical discernibility of formulas" (p. 163). 42 The dual counterpart of Peano arithmetic can be interpreted as a definable model of Peano arithmetic in a model of Peano arithmetic as this is described by Ikeda and Tsuboi (2007) as well as by Strannegård (1999), also by Murawski (1988) or even in Heyting arithmetic as a "constructive nonstandard model" (Moerdijk, Palmgren 1997). 43 The dual Peano arithmetic involved in Hilbert arithmetic can be also interpreted by means of the result provided by Enayat (2008).…”

Section: The Provability Of Flt In Hilbert Arithmeticmentioning

confidence: 99%

Fermat’s last theorem proved in Hilbert arithmetic. I. From the proof by induction to the viewpoint of Hilbert arithmetic

Пенчев

2021

SSRN Journal

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“…It follows from the fact that @ef(M), M, E ) I= A;)d 1 + C :-AC for any countable recursively saturated M C PA (cf. (21, [15] or Theorem 8 above) and TARSKI'S theorem. Similarly A;lA + C i-AC does not prove the existence of satisfaction classes for L" ( n E N, n > 0).…”

Section: Existence and Variety Of Iterations Of Satisfaction Classesmentioning

confidence: 99%

“…More exactly we shall study the problem of constructing models of AJdf + Cf-AC for a given nonstandard countable model M of PA. A similar problem was discussed in [15], where we considered models of A J A + Ci-AC which consists of families of sets definable by nonstandard formulas in the seme of a given satisfaction class. In the present paper we shall show how to generalize that construction by considering iterations of satisfaction classes.…”

Section: (4)mentioning

confidence: 99%