  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 72 (2021) 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 Szemer\'{e}di's Theorem can be carried
out in SPOTS, which is conservative over ZF + ACC (the Axiom of Countable
Choice). The theory SCOTS is a conservative extension of ZF + ADC.
