Given a resplendent model for Peano arithmetic there exists a full satisfaction class over , i.e. an assignment of truth-values, to all closed formulas in the sense of with parameters from , which satisfies the usual semantic rules. The construction is based on the consistency of an appropriate system of -logic which is proved by an analysis of standard approximations of nonstandard formulas.
Zeitseiir. j . iIi(bth. Logik titid Grutzdlugeii d. X a l h . lid. JZ, S . SJl-624 ( 1 8 8 6 ) H. KOTLARSKI T h e o r e m 1.4. Kaufmann's model M admits no full satisfaction class (cf. SCHMERL [12] RATAJCZYK [lo] studied satisfaction classes from another point of view. He gave a combinatorial axiomatisation of the arithmetical part of the theory PA(#) = PA + S is a full satisfaction class + induction in L,, u {a}, and gave an independent sentence in the Paris-Harrington style.It should be noticed that a weaker notion, t,hat of a partial satisfaction class was applied to study recursively saturated models of PA, see SCHMERL [12], KIRBY, MCALOON and MURAWSKI [l] and KOTLARSKI [4].The notion of a full satisfaction class is a single sentence in L,, u ( 8 ) . The aim of this paper is to study models of the theory do-PA(#) = PA + S is a full satisfaction class + do-induction in L,,u ( S } .We hope that a t least some of the ideas developed below ill shed some light on problems about do-PA, see PARIS-WILKIE [9].for the definition and construction of a Kaufmann model).
Finite axiomatisability
I n this section we shall prove T h e o r e m 2.1. Ao-PA(S) is finitely axiomatisable.Let PAbe the theory in L,, consisting of recursive definitions of addition and multiplication and the definition of inequality. It is well known that PAis sufficiently strong to represent all the recursive functions (SHOENFIELD [13]), FO to arithmetise the language. Thus PAallows us to state the definition of a full satisfaction class.Let T be the following theory:
We give some information about the action of A u t ( M ) on M(O), where M is a countable arithmetically saturated model of Peano Arithmetic. We concentrate on analogues of moving gaps and covering gaps inside M ( 0 ) . Mathematics Subject Classification: 03C62, 03C50, 03H15.
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