# SAS

## Seminar

### Nonstandard Analysis: a new way to compute

Seminar Room 1, Newton Institute

#### Abstract

Constructive 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.

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