Ideas from 'works' by John von Neumann [1935], by Theme Structure
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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

4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
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

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




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.
