Combining Philosophers

Ideas for Bryan van Norden, Andy Douglas and John von Neumann

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3 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers

13489	Von Neumann treated cardinals as a special sort of ordinal [Neumann, by Hart,WD]
	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]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers

22716	Von Neumann defined ordinals as the set of all smaller ordinals [Neumann, by Poundstone]
	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.

12336	A von Neumann ordinal is a transitive set with transitive elements [Neumann, by Badiou]
	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