Abstract
Model-theoretic frameworks for Nonstandard Analysis depend on the existence of nonprincipal ultrafilters, a strong form of the Axiom of Choice (AC). Hrbacek and Katz, APAL Volume 72 formulate axiomatic nonstandard set theories SPOT and SCOT that are conservative extensions of respectively ZF and ZF + ADC (the Axiom of Dependent Choice), and in which a significant part of Nonstandard Analysis can be developed.The present paper extends these theories to theories with many levels of standardness, called respectively SPOTS and SCOTS. It shows that Jin's recent nonstandard proof of Szemeredi's Theorem can be carried out in SPOTS and that SCOTS is a conservative extension of ZF + ADC.
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