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Ideas of John von Neumann, by Text

[Hungarian, 1903 - 1957, One of the fathers of the modern computer.]

1923 On the Introduction of Transfinite Numbers
p.24 Von Neumann numbers are preferred, because they continue into the transfinite
     Full Idea: Von Neumann's version of the natural numbers is in fact preferred because it carries over directly to the transfinite ordinals.
     From: comment on John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n9
p.24 For Von Neumann the successor of n is n U {n} (rather than {n})
     Full Idea: For Von Neumann the successor of n is n U {n} (rather than Zermelo's successor, which is {n}).
     From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n8
p.73 Von Neumann treated cardinals as a special sort of ordinal
     Full Idea: Von Neumann's decision was to start with the ordinals and to treat cardinals as a special sort of ordinal.
     From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by William D. Hart - The Evolution of Logic 3
     A reaction: [see Hart 73-74 for an explication of this]
p.122 Each Von Neumann ordinal number is the set of its predecessors
     Full Idea: Each Von Neumann ordinal number is the set of its predecessors. ...He had shown how to introduce ordinal numbers as sets, making it possible to use them without leaving the domain of sets.
     From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Shaughan Lavine - Understanding the Infinite V.3
p.128 A von Neumann ordinal is a transitive set with transitive elements
     Full Idea: In Von Neumann's definition an ordinal is a transitive set in which all of the elements are transitive.
     From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Alain Badiou - Briefings on Existence 11
1925 An Axiomatization of Set Theory
p.209 All the axioms for mathematics presuppose set theory
     Full Idea: There is no axiom system for mathematics, geometry, and so forth that does not presuppose set theory.
     From: John von Neumann (An Axiomatization of Set Theory [1925]), quoted by Stewart Shapiro - Foundations without Foundationalism 8.2
     A reaction: Von Neumann was doubting whether set theory could have axioms, and hence the whole project is doomed, and we face relativism about such things. His ally was Skolem in this.
p.215 Limitation of Size is not self-evident, and seems too strong
     Full Idea: Von Neumann's Limitation of Size axiom is not self-evident, and he himself admitted that it seemed too strong.
     From: comment on John von Neumann (An Axiomatization of Set Theory [1925]) by Shaughan Lavine - Understanding the Infinite VII.1
1935 works
p.188 Von Neumann wanted mathematical functions to replace sets
     Full Idea: Von Neumann suggested that functions be pressed into service to replace sets.
     From: report of John von Neumann (works [1935]) by Josť A. Benardete - Metaphysics: the logical approach Ch.23
p.280 Von Neumann defines each number as the set of all smaller numbers
     Full Idea: Von Neumann defines each number as the set of all smaller numbers.
     From: report of John von Neumann (works [1935]) by Simon Blackburn - Oxford Dictionary of Philosophy p.280