green numbers give full details.     |    back to list of philosophers     |     unexpand these ideas

### 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 18180 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 ) by Penelope Maddy - Naturalism in Mathematics I.2 n9
 p.24 18179 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 ) by Penelope Maddy - Naturalism in Mathematics I.2 n8
 p.73 13489 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 ) by William D. Hart - The Evolution of Logic 3 A reaction: [see Hart 73-74 for an explication of this]
 p.122 15925 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 ) by Shaughan Lavine - Understanding the Infinite V.3
 p.128 12336 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 ) by Alain Badiou - Briefings on Existence 11
 1925 An Axiomatization of Set Theory
 p.209 13672 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 ), 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 15943 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 ) by Shaughan Lavine - Understanding the Infinite VII.1
 1935 works
 p.188 3355 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 ) by José A. Benardete - Metaphysics: the logical approach Ch.23
 p.280 3340 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 ) by Simon Blackburn - Oxford Dictionary of Philosophy p.280