Omega‐ and Beta‐Models of Alternative Set Theory

Abstract

AbstractWe present the axioms of Alternative Set Theory (AST) in the language of second‐order arithmetic and study its ω‐ and β‐models. These are expansions of the form (M, M), M ⊆ P(M), of nonstandard models M of Peano arithmetic (PA) such that (M, M) ⊩ AST and ω ϵ M. Our main results are: (1) A countable M ⊩ PA is β‐expandable iff there is a regular well‐ordering for M. (2) Every countable β‐model can be elementarily extended to an ω‐model which is not a β‐model. (3) The Ω‐orderings of an ω‐model (M, M) are absolute well‐orderings iff the standard system SS(M) of M is a β‐model of A−2. (4) There are ω‐expandable models M such that no ω‐expansion of M contains absolute Ω‐orderings. (5) There are s‐expandable models (i. e., their ω‐expansions contain only absolute Ω‐orderings) which are not β‐expandable. (6) For every countable β‐expansion M of M, there is a generic extension M[G] which is also a β‐expansion of M. (7) If M is countable and β‐expandable, then there are regular orderings <1, <2 such that neither <1 belongs to the ramified analytical hierarchy of the structure (M, <2), nor <2 to that of (M, <1). (8) The result (1) can be improved as follows: A countable M ⊩ PA is β‐expandable iff there is a semi‐regular well‐ordering for M.