# Nonstandard Analysis: a new way to compute

Sam Sanders ( University of Ghent )

- 14:00 17th May 2013 ( week 4, Trinity Term 2013 )

Following recent results in Reverse Mathematics, we introduce

"$\Omega$-invariance", a simple and elegant notion from Nonstandard Analysis

meant to capture the notion of algorithm (both in the sense of Alan Turing and

Errett Bishop). Intuitively, an object is $\Omega$-invariant if it does not depend

on the

**choice**of infinitesimal used in its definition.We show that $\Omega$-invariance exactly captures the notion of Turing

computability and we identify an extra condition T (based on the Transfer Principle

of Nonstandard Analysis) so that "$\Omega$-invariance + condition T " captures

the

**constructive**notion of algorithm à la Bishop.