Multi-level Nonstandard Analysis and the Axiom of Choice

TL;DR

This paper extends axiomatic nonstandard set theories to multiple levels of standardness, enabling the development of Nonstandard Analysis without relying on the full Axiom of Choice, and demonstrates its application to Szemerédi's Theorem.

Contribution

It introduces multi-level nonstandard set theories SPOTS and SCOTS, generalizing previous frameworks and showing their utility in nonstandard proofs of combinatorial theorems.

Findings

SPOTS supports Jin's nonstandard proof of Szemerédi's Theorem.

SCOTS is a conservative extension of ZF + ADC.

Theories avoid reliance on nonprincipal ultrafilters.

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 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. The theory SCOTS is a conservative extension of ZF + ADC.