Diagonal Arguments

0483331 - FLÚ 2018 RIV CZ eng J - Journal Article
Peregrin, Jaroslav
Diagonal Arguments.
Acta Universitatis Carolinae. Philosophica et Historica. -, č. 2 (2017), s. 33-43. ISSN 0567-8293
R&D Projects: GA ČR(CZ) GA17-15645S
Institutional support: RVO:67985955
Keywords : diagonalization * cardinality * Russell’s paradox * incompleteness of arithmetic
OECD category: Philosophy, History and Philosophy of science and technology

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.
Permanent Link: http://hdl.handle.net/11104/0278685