Omega- and Beta-Models of Alternative Set Theory

We 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 ? 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 MG] 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.