✍ Scribed by Roman Murawski
No coin nor oath required. For personal study only.
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
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, S) of subsets of M as follows:
x ~D e f d M , S ) ifT ( E P ) ~( E P ) ~( M , S ) + (Frn(cp)&(x)(x€X= S(P,;X,P))),
(q I% v ) E I, ((Euk) E I.
l ) The author thanks H. K ~S K I for helpful discussions and suggestions.
📜 SIMILAR VOLUMES
In this study, artificial neural network (ANN) and fuzzy logic (FL) models have been developed for predicting the compressive strength (f c ) and dynamic modulus of elasticity (E d ) of the crushed tile concretes (CTC) exposed to elevated temperatures. Some relationships are established between chos