Nonstandard Analysis: a new way to compute
Seminar Room 1, Newton Institute
AbstractConstructive Analysis was introduced by Errett Bishop to identify the `computational meaning' of mathematics. Bishop redeveloped mathematics, in the spirit of intuitionistic mathematics, based on primitive notions like algorithm, explicit computation, and finite procedure. The exact meaning of these vague terms was left open, to ensure the compatibility of Constructive Analysis with several traditions in mathematics. Constructive Reverse Mathematics (CRM) is a spin-off from Harvey Friedman's famous Reverse Mathematics program, based on Constructive Analysis.
In this talk, we introduce `$\Omega$-invariance': a simple and elegant definition of finite procedure in (classical) Nonstandard Analysis. We show that $\Omega$-invariance captures Bishop's notion of algorithm quite well. In particular, using an intuitive interpretation based on $\Omega$-invariance, we obtain many results from CRM inside Nonstandard Analysis. Similar results for Computability (aka Recursion) Theory are also discussed.
This research is made possible through the generous support of a grant from the John Templeton Foundation for the project Philosophical Frontiers in Reverse Mathematics. Please note that the opinions expressed in this publication are those of the author and do not necessarily reflect the views of the John Templeton Foundation.