Internal approach to external sets and universes

Abstract

A problem which enthusiasts of IST, Nelson's internal set theory, usually face is how to treat external sets in the internal universe which does not contain them directly. To solve this problem, we consider BST, bounded set theory, a modification of IST which is, briefly, a theory for the family of those IST sets which are members of standard sets. We show that BST is strong enough to incorporate external sets in the internal universe in a way sufficient to develop the most advanced applications of nonstandard methods. In particular, we define in BST an enlargement of the BST universe which satisfies the axioms of HST, an external theory close to a theory introduced by Hrbaeek. HST includes Replacement and Saturation for all formulas but contradicts the Power Set and Choice axioms (either of them is incompatible with Replacement plus Saturation), therefore to get an external universe which satisfies all of ZFC minus Regularity one has to pay by a restriction of Saturation. We prove that HST admits a system of subuniverses which model ZFC (minus Regularity but with Power Set and Choice) and Saturation in a form restricted by a fixed but arbitrary standard cardinal. Thus the proposed system of set theoretic foundations for nonstandard mathematics, based on the simple and natural axioms of the internal theory BST, provides the treat- ment of external sets sufficient to carry out elaborate external constructions. This article t is the first in the series of three articles devoted to set theoretic