Ideas of John von Neumann, by Theme

[Hungarian, 1903 - 1957, One of the fathers of the modern computer.]

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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
Von Neumann defines each number as the set of all smaller numbers [Blackburn]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Limitation of Size is not self-evident, and seems too strong [Lavine]
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
Von Neumann wanted mathematical functions to replace sets [Benardete,JA]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
Von Neumann treated cardinals as a special sort of ordinal [Hart,WD]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
A von Neumann ordinal is a transitive set with transitive elements [Badiou]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / g. Von Neumann numbers
For Von Neumann the successor of n is n U {n} (rather than {n}) [Maddy]
Von Neumann numbers are preferred, because they continue into the transfinite [Maddy]
Each Von Neumann ordinal number is the set of its predecessors [Lavine]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
All the axioms for mathematics presuppose set theory