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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.
Gist of Idea
Von Neumann defined ordinals as the set of all smaller ordinals
Source
report of John von Neumann (works [1935]) by William Poundstone - Prisoner's Dilemma 02 'Sturm'
Book Ref
Poundstone,William: 'Prisoner's Dilemma' [OUP 1992], p.29
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.
| 15943 | Limitation of Size is not self-evident, and seems too strong [Lavine on Neumann] |
| 13672 | All the axioms for mathematics presuppose set theory [Neumann] |
| 12336 | A von Neumann ordinal is a transitive set with transitive elements [Neumann, by Badiou] |
| 13489 | Von Neumann treated cardinals as a special sort of ordinal [Neumann, by Hart,WD] |
| 18180 | Von Neumann numbers are preferred, because they continue into the transfinite [Maddy on Neumann] |
| 18179 | For Von Neumann the successor of n is n U {n} (rather than {n}) [Neumann, by Maddy] |
| 15925 | Each Von Neumann ordinal number is the set of its predecessors [Neumann, by Lavine] |
| 3340 | Von Neumann defines each number as the set of all smaller numbers [Neumann, by Blackburn] |
| 3355 | Von Neumann wanted mathematical functions to replace sets [Neumann, by Benardete,JA] |
| 22716 | Von Neumann defined ordinals as the set of all smaller ordinals [Neumann, by Poundstone] |