Penchev, Vasil (2022) Gödel mathematics versus Hilbert mathematics. I The Gödel incompleteness (1931) statement: axiom or theorem? [Preprint]
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GÖDEL MATHEMATICS VERSUS HILBERT MATHEMATICS.pdf Download (590kB) | Preview |
Abstract
The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: weather it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic. The main argument consists in the contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. Thus, the pair of arithmetic and set are similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate: correspondingly, by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. The axiom of choice transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. The Gödel incompleteness statement relies on the contradiction of the axioma of induction and infinity.
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| Item Type: | Preprint | ||||||
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| Keywords: | Boolean algebra, completeness, dual axiomatics, Euclidean and non-Euclidean geometries, Gödel, the Fifth postulate, finitism, foundations of mathematics, Hilbert arithmetic, Hilbert program, Husserl, incompleteness, logicism, Peano arithmetic, phenomenology, Principia mathematica, propositional logic, Riemann’s “space curvature”, Russell, set theory | ||||||
| Subjects: | Specific Sciences > Mathematics > Logic General Issues > History of Philosophy of Science General Issues > History of Science Case Studies |
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| Depositing User: | Prof. Vasil Penchev | ||||||
| Date Deposited: | 19 Oct 2022 17:13 | ||||||
| Last Modified: | 19 Oct 2022 17:13 | ||||||
| Item ID: | 21280 | ||||||
| Official URL: | https://www.cambridge.org/engage/coe/article-detai... | ||||||
| DOI or Unique Handle: | https://doi.org/10.33774/coe-2022-wlr02 | ||||||
| Subjects: | Specific Sciences > Mathematics > Logic General Issues > History of Philosophy of Science General Issues > History of Science Case Studies |
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| Date: | 17 October 2022 | ||||||
| URI: | http://philsci-archive.pitt.edu/id/eprint/21280 |
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