# The unreasonable effectiveness of Nonstandard Analysis

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Mathematical, Foundational and Computational Aspects of the Higher Infinite

The aim of my talk is to highlight a hitherto unknown computational aspect of Nonstandard Analysis. In particular, we provide an algorithm which takes as input the proof of a mathematical theorem from pure Nonstandard Analysis, i.e. formulated solely with the nonstandard definitions (of continuity, integration, dif- ferentiability, convergence, compactness, et cetera), and outputs a proof of the as- sociated effective version of the theorem. Intuitively speaking, the effective version of a mathematical theorem is obtained by replacing all its existential quantifiers by functionals computing (in a specific technical sense) the objects claimed to exist. Our algorithm often produces theorems of Bishops Constructive Analysis (). The framework for our algorithm is Nelsons syntactic approach to Nonstandard Analysis, called internal set theory (), and its fragments based on Goedels T as introduced in . Finally, we establish that a theorem of Nonstandard Analysis has the same computational content as its highly constructive Herbrandisation. Thus, we establish an algorithmic two-way street between so-called hard and soft analysis, i.e. between the worlds of numerical and qualitative results.

References:  Benno van den Berg, Eyvind Briseid, and Pavol Safarik, A functional interpretation for non- standard arithmetic, Ann. Pure Appl. Logic 163 (2012), no. 12, 19621994.

 Errett Bishop and Douglas S. Bridges, Constructive analysis, Grundlehren der Mathematis- chen Wissenschaften, vol. 279, Springer-Verlag, Berlin, 1985.

 Fernando Ferreira and Jaime Gaspar, Nonstandardness and the bounded functional interpre- tation, Ann. Pure Appl. Logic 166 (2015), no. 6, 701712.

 Edward Nelson, Internal set theory: a new approach to nonstandard analysis, Bull. Amer. Math. Soc. 83 (1977), no. 6, 11651198.

 Stephen G. Simpson, Subsystems of second order arithmetic, 2nd ed., Perspectives in Logic, CUP , 2009.

This talk is part of the Isaac Newton Institute Seminar Series series.

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