December 12
Karel Hrbáček,
CUNY
Multi-level nonstandard analysis, the axiom of choice, and recent work of R. Jin
Model-theoretic frameworks for nonstandard methods require the existence of nonprincipal ultrafilters over N, a strong form of the Axiom of Choice (AC). While AC is instrumental in many abstract areas of mathematics, its use in infinitesimal calculus or number theory should not be necessary.
In the paper KH and M. G. Katz, Infinitesimal analysis without the Axiom of Choice, Ann. Pure Applied Logic 172, 6 (2021), https://arxiv.org/abs/2009.04980, we have formulated SPOT, a theory in the language that has, in addition to membership, a unary predicate 'is standard.' The theory extends ZF by three simple axioms, Transfer, Nontriviality and Standard Part, that reflect the insights of Leibniz. It is a subtheory of the nonstandard set theories IST and HST, but unlike them, it is a conservative extension of ZF. Arguments carried out in SPOT thus do not depend on any form of AC. Infinitesimal calculus can be developed in SPOT. A stronger theory SCOT is a conservative extension of ZF + Dependent Choice. It is suitable for handling such features as an infinitesimal approach to the Lebesgue measure.
Renling Jin recently gave a groundbreaking nonstandard proof of Szemeredi's theorem in a model-theoretic framework that has three levels of infinity. I will formulate and motivate SPOTS, a multi-level version of SPOT, carry out Jin's proof of Ramsey's theorem in SPOTS, and discuss how his proof of Szemeredi's theorem can be developed in it.
While it is still open whether SPOTS is conservative over ZF, SCOTS (the multi-level version of SCOT) is a conservative extension of ZF + Dependent Choice.