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Diagonal Arguments
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SYSNO ASEP 0483331 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Ostatní články Title Diagonal Arguments Author(s) Peregrin, Jaroslav (FLU-F) RID, ORCID, SAI Source Title Acta Universitatis Carolinae. Philosophica et Historica. - : Nakladatelství Karolinum - ISSN 0567-8293
-, č. 2 (2017), s. 33-43Number of pages 20 s. Publication form Print - P Language eng - English Country CZ - Czech Republic Keywords diagonalization ; cardinality ; Russell’s paradox ; incompleteness of arithmetic Subject RIV AA - Philosophy ; Religion OECD category Philosophy, History and Philosophy of science and technology R&D Projects GA17-15645S GA ČR - Czech Science Foundation (CSF) Institutional support FLU-F - RVO:67985955 DOI 10.14712/24647055.2017.14 Annotation It is a trivial fact that if we have a square table filled with numbers, we can always form a column which is not yet contained in the table. Despite its apparent triviality, this fact can lead us the most of the path-breaking results of logic in the second half of the nineteenth and the first half of the twentieth century. We explain how this fact can be used to show that there are more sequences of natural numbers than there are natural numbers, that there are more real numbers than natural numbers and that every set has more subsets than elements (all results due to Cantor), we indicate how this fact can be seen as underlying the celebrated Russell’s paradox, and we show how it can be employed to expose the most fundamental result of mathematical logic of the twentieth century, Gödel’s incompleteness theorem. Finally, we show how this fact yields the unsolvability of the halting problem for Turing machines. Workplace Institute of Philosophy Contact Chlumská Simona, chlumska@flu.cas.cz ; Tichá Zuzana, asep@flu.cas.cz Tel: 221 183 360 Year of Publishing 2018
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