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The Recursively Saturated Part of Models of Peano Arithmetic

✍ Scribed by Henryk Kotlarski

Publisher
John Wiley and Sons
Year
1986
Tongue
English
Weight
403 KB
Volume
32
Category
Article
ISSN
0044-3050

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✦ Synopsis

PA we define the rfcursively saturated part of XU by RS(9Jl) = ( a E $1: ( 3 8 < YJ?) ( a E )%I and 8 is recursively saturated)).

We shall study various possibilities for the relationship between 912 and RS(XU). Tliii paper has grown out of our observation that it may happen that , D is a simple extension (of the minimal model of Th?n) and has a non-empty recursively saturated part: this is implicit in SMORY~SKI-STAVI [6] and KOTLARSKI [3]. Several ideas used in this paper are taken from SMORY~SKI-STAVI [6] and KOTLARSKI [4].

Our first observation is the following T h e o r e m 1. For any XU 1 PA, RS( 93) is either empty 0 1 it i s an elementary initial segmrd of 9Jl.

Proof. Suppose that a , , . . ., a, E 1ER1. We may assume that a , < . . . < a,. Pick Y2, < W, i = 1, . . . . n, such that a, E and %, is recursively saturated. Extend Y?, , to 9i, = [b E 19Rl: ( 3 c E I%,l) b 5 c} By GAIFMAN [l, proposition 2.21, 8, <En. By SMORT~SKI-STAVI [6], 8, is recursively saturated. It follows that if t(v,, . . . . a,) is any trrm of the language of PA then t(a,,

, a,) E 1!JlnJ, therefore RS(YJ?) is an elementary submodel of XU (as it is closed under Skolem functions). Moreover, if h 5 c, then c E RS(93), and exactly the same argument gives that b E RS(CrR).

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