The Diversity of Minimal Cofinal Extensions

Abstract

Fix a countable nonstandard model M of Peano Arithmetic. Even with some rather severe restrictions placed on the types of minimal cofinal extensions N ≻ M that are allowed, we still find that there are 20 possible theories of (N ,M) for such N ’s. The script letters M,N ,K (possibly adorned) always denote models of Peano Arithmetic (PA) having domains M,N,K, respectively. The set of parametrically definable subsets of M is Def(M). If J ⊆ M , then Cod(M/J) = {A ∩ J : A ∈ Def(M)}. A cut of M is a subset J ⊆ M such that 0 ∈ J 6= M and if a ≤ b ∈ J , then a + 1 ∈ J . The cut J is exponentially closed if 2 ∈ J whenever a ∈ J . Suppose that M ≺ N . Their Greatest Common Initial Segment is GCIS(M,N ) = {b ∈ M : whenever N |= a ≤ b, then a ∈ M}, which is M if N is an end extension of M and is a cut otherwise. If J is a cut of M, then N fills J if there is b ∈ N such that whenever a ∈ J and c ∈ M\J , then N |= a < b < c. The interstructure lattice is Lt(N /M) = {K : M 4 K 4 N}, ordered by elementary extension. If 1 ≤ n < ω, then n is the lattice that is a chain of n elements. One of the themes of [4] is the diversity of cofinal extensions, exemplified by the following theorem. Theorem A: ([4, Theorem 7.1]) If J is an exponentially closed cut of countable M, then there is a set C of cofinal elementary extensions of M such that: (1) |C| = 20 ; (2) if N ∈ C, then GCIS(M,N ) = J , Cod(N /J) = Cod(M/J) and N does not fill J ; (3) if N1,N2 ∈ C are distinct, then Th(N1,M) 6= Th(N2,M); (4) Lt(N /M) ∼= 3 for each N ∈ C. ([4, page 285]) It was left open, and specifically asked ([4, Question 7.5]), whether the 3 in (4) can be replaced by 2 (so that every N ∈ C is a minimal Date: September 17, 2021.