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Nonstandard Analysis: a new way to compute

Sam Sanders ( University of Ghent )

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.  

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