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Model-theoretic frameworks for nonstandard methods entail the existence of nonprincipal ultrafilters over $\mathbb{N}$, a strong version of the Axiom of Choice ($\mathbf{AC}$). While $\mathbf{AC}$ is instrumental in many abstract areas of mathematics, such as general topology or functional analysis, its use in infinitesimal calculus or number theory should not be necessary.

In [1], Mikhail Katz and I have formulated a set theory $\mathbf{SPOT}$ in the $\mathbf{st}$-$\in$-language. The theory has three simple axioms, Transfer, Nontriviality and Standard Part. It is a subtheory of the nonstandard set theories $\mathbf{IST}$ and $\mathbf{HST}$, but unlike them, it is a conservative extension of $\mathbf{ZF}$. Arguments carried out in $\mathbf{SPOT}$ thus do not depend on any form of $\mathbf{AC}$. Infinitesimal calculus can be developed in $\mathbf{SPOT}$ as far as the global version of Peano's Theorem (the usual proofs of which use $\mathbf{ADC}$, the Axiom of Dependent Choice). The existence of upper Banach densities can be proved in $\mathbf{SPOT}$. The conservativity of $\mathbf{SPOT}$ over $\mathbf{ZF}$ is established by a construction that combines and extends the methods of forcing developed by A. Enayat and M. Spector.

A stronger theory $\mathbf{SCOT}$ is a conservative extension of $\mathbf{ZF} + \mathbf{ADC}$. It is suitable for handling such features as an infinitesimal approach to the Lebesgue measure.

I will also explore the possibilities for extending these theories to multiple levels of standardness and the relevance of such extensions to infinitesimal calculus and R. Jin's proof of Szemerédi's Theorem.

[1] K. Hrbacek and M. G. Katz, Infinitesimal analysis without the Axiom of Choice, Ann. Pure Appl. Logic 172, 6 (2021).

https://doi.org/10.1016/j.apal.2021.102959

https://arxiv.org/abs/2009.04980