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Bernays and the completeness theorem
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Dean, Walter (2017) Bernays and the completeness theorem. Annals of the Japan Association for Philosophy of Science, 25 . pp. 45-55.
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Official URL: http://phsc.jp/en/annals.html
Abstract
A well-known result in Reverse Mathematics is the equivalence of the formalized version of the Gödel completeness theorem [8] – i.e. every countable, consistent set of first-order sentences has a model – and Weak König's Lemma [WKL] – i.e. every infinite tree of 0-1 sequences contains an infinite path– over the base theory RCA0. It is less well known how the Completeness Theorem came to be studied in the setting of second-order arithmetic and computability theory. The first goal of this note will be to recount these developments against the backdrop of the latter phases of the Hilbert program, culminating in the publication of the second volume of Hilbert and Bernays’s [13] Grundlagen der Mathematiks in 1939. This work contains a detailed formalization of the Completeness Theorem in a system similar to first-order Peano arithmetic [PA] – a result which has come to be known as the Arithmetized Completeness Theorem. Its second goal will be to illustrate how reflection on this result informed Bernays’s views about the philosophy of mathematics, in particular in regard to his engagement with the maxim “consistency implies existence”.
| Item Type: | Journal Article | ||||||
|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Social Sciences > Philosophy | ||||||
| Library of Congress Subject Headings (LCSH): | Completeness theorem, Gödel's theorem, Hilbert, David, 1862-1943 -- Influence, Bernays, Paul, 1888-1977 -- Influence | ||||||
| Journal or Publication Title: | Annals of the Japan Association for Philosophy of Science | ||||||
| Publisher: | Kagaku Kisoron Gakkai | ||||||
| ISSN: | 0453-0691 | ||||||
| Official Date: | 2017 | ||||||
| Dates: |
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| Date of first compliant deposit: | 20 December 2016 | ||||||
| Volume: | 25 | ||||||
| Page Range: | pp. 45-55 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Related URLs: |
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