Eastaugh, Benedict (2018) Computational reverse mathematics and foundational analysis. [Preprint]

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Abstract
Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, nonsettheoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account can be fruitfully applied in the evaluation of major foundational approaches by a careful examination of two case studies: a partial realization of Hilbert’s program due to Simpson [1988], and predicativism in the extended form due to Feferman and Schütte.
Shore [2010, 2013] proposes that equivalences in reverse mathematics be proved in the same way as inequivalences, namely by considering only omegamodels of the systems in question. Shore refers to this approach as computational reverse mathematics. This paper shows that despite some attractive features, computational reverse mathematics is inappropriate for foundational analysis, for two major reasons. Firstly, the computable entailment relation employed in computational reverse mathematics does not preserve justification for the foundational programs above. Secondly, computable entailment is a Pi11 complete relation, and hence employing it commits one to theoretical resources which outstrip those available within any foundational approach that is prooftheoretically weaker than Pi11CA0.
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Item Type:  Preprint  

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Keywords:  reverse mathematics, foundations of mathematics  
Subjects:  Specific Sciences > Mathematics > Epistemology Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Logic Specific Sciences > Mathematics > Proof Specific Sciences > Mathematics 

Depositing User:  Benedict Eastaugh  
Date Deposited:  30 Jun 2018 12:34  
Last Modified:  30 Jun 2018 12:34  
Item ID:  14825  
Subjects:  Specific Sciences > Mathematics > Epistemology Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Logic Specific Sciences > Mathematics > Proof Specific Sciences > Mathematics 

Date:  28 June 2018  
URI:  https://philsciarchive.pitt.edu/id/eprint/14825 
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