February 27
Karel Hrbáček,
CUNY
Theory of Infinitesimals
In 1961 Abraham Robinson solved a centuries-old problem by developing rigorous foundations for infinitesimal calculus. His model-theoretic approach was criticized for its dependence on the axiom of choice and its lack of categoricity. I will argue that the axiomatic approach can overcome these objections.
Starting with the ideas that can be found in the writings of Leibniz and other early infinitesimalists, I will present the theory SPOT, a conservative extension of ZF, that is capable of developing elementary analysis via infinitesimals. Natural generalizations then lead to theories that enable techniques covering almost the entire spectrum of nonstandard analysis. The final theory in the sequence, BST, is 'categorical over ZFC.' Similar results are obtained for theories with multiple levels of standardness. A further extension of the language of these theories allows for a simple presentation of recent results of R. Jin and M. Di Nasso; I will give Jin’s proof of Ramsey’s theorem as an example.
The set-theoretic view of the Leibnizian continuum presents a challenge to traditional set theory, as the existence of infinitesimals entails the existence of unlimited ('infinite') natural numbers. I will indicate how the above theories can be formulated from an 'external' point of view, in terms of an embedding of the standard universe into the internal universe.
This is joint work with Mikhail G. Katz.