The Isomorphism Property Versus the Special Model Axiom

This paper answers some questions of D. Ross in [R]. In x1, we show that some consequences of the @0 or @1{special model axiom in [R] can not be proved by the {isomorphism property for any cardinal . In x2, we show that with one exception, the @0{isomorphism property does imply the remaining consequences of the special model axiom in [R]. In x3, we improve a result in [R] by showing that the {special model axiom is equivalent to the @0{special model axiom plus {saturation. x0. Notation and introduction Throughout this paper we use ; ; : : : for in nite cardinals, ; ; ; : : : for ordinals, L;L; : : : for some rst{order languages and A;B; : : : for models (or structures) with base sets (or universes) A;B; : : :. Let N be the set of all natural numbers. By a standard universe we mean the superstructure V!(N) = S n2! Vn with the \2" relation, where V0 = N , a set of urelements, and Vn+1 = Vn S P(Vn). jV!(N)j = i!. By a nonstandard universe we mean the image of V under Mostowski collapse, where V is an elementary extension of the standard universe truncated at 2{rank !. We use V; V ; V ; V , etc. for standard or nonstandard universes. Note that each V contains a base set and a binary relation 2 . We will not distinguish a nonstandard universe from its base set. We let N (R) be all natural (real) numbers and N (R) be all standard and nonstandard natural (real) numbers in V . If V is not explicitly given, we usually use N ( R) instead of N (R). If P and Q are two linear orders, an order{preserving map f : P 7 ! Q is called a co nal (coinitial) embedding if f [P ] is upper (lower) unbounded in Q. cf(Q) (ci(Q)) is the least cardinal such that (the reverse order of ) can be co nally (coinitially) embedded into Q. We let cf( N) mean the co nality of N with the usual order and ci( N) mean the coinitiality of N N with the usual order. A set A is called internal in V if A is an element of V . If V is not explicitly given, A is internal means A is an element of a nonstandard universe. An L{structure A is