We present an axiomatic framework for nonstandard analysis-the Nonstandard Class Theory which extends von Neumann-Godel-Bernays Set Theory by adding a unary predicate symbol St to the language of NBG means that the class X is standard) and axioms-related to it- analogs of Nelson's idealization, standardization and transfer principles. Those principles are formulated as axioms, rather than axiom schemes, so that NCT is finitely axiomatizable. NCT can be considered as a theory of definable classes of Bounded Set Theory by V. Kanovei (...) and M. Reeken. In many aspects NCT resembles the Alternative Set Theory by P. Vopenka. For example there exist semisets in NCT and it can be proved that a set has a standard finite cardinality iff it does not contain any proper subsemiset. Semisets can be considered as external classes in NCT. Thus the saturation principle can be formalized in NCT. (shrink)

We develop an axiomatic set theory — the Theory of Hyperfinite Sets THS— which is based on the idea of the existence of proper subclasses of large finite sets. We demonstrate how theorems of classical continuous mathematics can be transfered to THS, prove consistency of THS, and present some applications.