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Computational reverse mathematics and foundational analysis

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, non-set-theoretic 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 omega-models 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 Pi-1-1 complete relation, and hence employing it commits one to theoretical resources which outstrip those available within any foundational approach that is proof-theoretically weaker than Pi-1-1-CA0.


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Item Type: Preprint
Creators:
CreatorsEmailORCID
Eastaugh, Benedictbenedict@eastaugh.net0000-0002-6629-3032
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: http://philsci-archive.pitt.edu/id/eprint/14825

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