Some Remarks on the Structure of Expansions

Abstract

in Poznali (Poland) l ) 0. Peano arithmetic P, as it was shown by K. GODEL, is not complete. Hence it is natural to investigate its extensions. We shall be interested in extensions "generated" by certain second order theories, namely by second order arithmetics T called' simply analysis. The question arises : what can be proved in T about natural number, i.e. what is the theoryThe semantical counterpart of this problem is the following: Call a model 93 I. P T-expandable iff i t is a number-theoretic part of a model of T . It turns out (cf. [I21 part 11, Theorem 37) that YJl t= PT iff there is a model !J? such that !J? 3 XV and % is T-expandable. Hence T-expandable models are good representatives of models of P".Investigating expandable models we can among others ask questions on the struchre of possible T-expansions of them. In this paper we shall study such a structure fur T being certain extensions of second order arithmetic A,. The main tool we shall use will be indicators. They were introduced by J. B. PARIS and L. A. S. KIRBY in [C;] (cf. also [51) in connection with the investigations of initial segments of countable nonstandard models of P having some combinatorial properties. We shall generalize the notion of indicator to the case of structures for second order arithmetic and then, using BARWISE'S approximation formulas for recursive close games, we shall build an indicator for the family of elementary initial segments (in the sense described below) of recursively saturated models of certain analyses. This will enable us to get some informations on the structure of expansions of some nonstandard models for P.We can look at our study also in another way. Namely we can treat the investigation of the structure of possible T-expansions of a nonstandard model \W t= P as t,he investigation of submodels (having the same number-theoretic part YJl ) of a given model 2l t= T. Hence we shall obtain also in our paper some informations on elementary submodels of countable recursively saturated non-o-models of certain analyses. I would like to express here my gratitude to Dr. Z. RATAJCZYK for helpful conversations and discussions.1. By Peano arithmetic we mean the first order theory formalized in the language with 0, 8, +, . and based on the usual axioms for successor, addition and multiplicstion and the axiom scheme of induction (cf. e.g.[IS]).The second order arithmetic A , (cf. for details e.g.[ 2 ] or [IS]) is a theory formulated in a first order language L(A;) extending the language L(P) of Peano arithmetic and containing two additionaI predicates: N ( x ) ("to be a naturaI number") and E (the 1) This is a part of author's doctoral dissertation writt,en under the supervision of W.MAREK and submitted to the University of Warsaw.