| 1923 | On the Introduction of Transfinite Numbers |
| 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 [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n8 |
| 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 [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n9 |
| 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 [1923]) 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 [1923]) 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 [1923]) 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 [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 | 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 [1925]) by Shaughan Lavine - Understanding the Infinite VII.1 |
| 1935 | works |
| p.29 | 22716 | Von Neumann defined ordinals as the set of all smaller ordinals | |
| Full Idea: At age twenty, Von Neumann devised the formal definition of ordinal numbers that is used today: an ordinal number is the set of all smaller ordinal numbers. | |||
| From: report of John von Neumann (works [1935]) by William Poundstone - Prisoner's Dilemma 02 'Sturm' | |||
| A reaction: I take this to be an example of an impredicative definition (not predicating something new), because it uses 'ordinal number' in the definition of ordinal number. I'm guessing the null set gets us started. |
| 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 [1935]) 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 [1935]) by Simon Blackburn - Oxford Dictionary of Philosophy p.280 |