Arithmetically Saturated Models of Arithmetic

Abstract

The paper presents an outline of the general theory of countable arithmetically saturated models of PA and some of its applications. We consider questions concerning the automorphism group of a countable recursively saturated model of PA. We prove new results concerning fixed point sets, open subgroups, and the cofinality of the automorphism group. We also prove that the standard system of a countable arithmetically saturated model of PA is determined by the lattice of its elementary substructures.is Th(N). The standard system of a model M , SSy(M ), is the family of those X ⊆ M for which there is an Y definable in M with parameters, such that X = ω ∩ Y. L PA will denote the language of PA.We will say that a type p( v, ā) in variables v = v 1 , . . . , v n , and parameters ā = a 1 , . . . , a m ∈ M is recursive, arithmetic, etc., if the set of Gödel numbers of formulas ϕ( v, w) ∈ p( v, w), where w = w 1 , . . . , w m , is recursive, arithmetic, etc. In the same sense we will speak of types as being subsets of ω.The notion of A-saturation was introduced by Wilmers in [16]. Let A be a family of subsets of ω. We say that a model M is A-saturated if the following two conditions are satisfied: (i) for every ā ∈ [M ] <ω , the type of ā, tp( ā), is in A; (ii) for every type p( v, ā) in A, if p( v, ā) is realized in some elementary extension of M , then it is realized in M .A Scott set is an ω-model of WKL 0 . The standard system of a model of PA is a Scott set; moreover, every countable Scott set is the standard system of a model of PA. If T is a completion of PA, X is a countable Scott set and T ∈ X , then there is a recursively saturated countable model M |= T such that SSy(M ) = X . Proofs of the above statements can be found in Kaye [2].The next proposition shows that a recursively saturated model of PA is much more than just recursively saturated.We will use a fixed arithmetical coding of finite sequences. If M is a model of PA and a, i ∈ M , then (a) i denotes the i-th term of the sequence coded by a, and lena is the length of the sequence coded by a.

Automorphism Groups of Countable Arithmetically Saturated Models of Peano Arithmetic

“…1.4(iii)], although it is equivalent. One direction of this equivalence is given in [13,Theorem 1.7(1b)]; the other is easy to see. It is straightforward to see that every recursively saturated model is uniformly ω-lofty.…”

“…This gave the first positive answer to the above question. A neater way, in which the use of the small index property is masked, that automorphism groups distinguish those models that are arithmetically saturated from the other countable recursively saturated ones was obtained the next year in [13,Coro. 3.9] (or see [15,Th.…”

“…Some special types of models of PA, the lofty models and those with the ω-property, are discussed in §2. In §3, we consider substructure lattices, carefully reviewing a result from [13]. Our main result, Theorem 4, is proved in §4.…”

Moving Intersticial Gaps

“…Earlier, in [9], it was proved that an arithmetically saturated M has an automorphism which moves every nondefinable element. It was asked in [9,Problem 4.5] if there is an automorphism g such that for any interstice Ω, either g(x) < x for all x ∈ Ω or g(x) > x for all x ∈ Ω. A consequence of Corollary 2.13 is that there is no such automorphism.…”

“…Two other proofs, perhaps a little more transparent than the original one, are in [9]. Suppose that g ∈ Aut(M) and a, b, d ∈ M are such that g(a) = a ≤ b < Gap(d).…”