Countable Models With Standard Part

Abstract

Publisher Summary This chapter discusses countable models. For a fixed language the class is not an elementary class because there is no set of axioms that would guarantee that N would be isomorphic to the natural numbers. However, it is well known that there is a finite set of axioms whose models all have an initial segment isomorphic to the natural numbers with their usual arithmetic and such that the recursive sets are numeral-wise definable. The initial segment of N is called standard numbers. The chapter discusses a unique countable model in a recursively axiomatized. It has been shown that a countable model of theory will be saturated if it contains an infinite set of indiscernibles.