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Full Idea
There is no axiom system for mathematics, geometry, and so forth that does not presuppose set theory.
Gist of Idea
All the axioms for mathematics presuppose set theory
Source
John von Neumann (An Axiomatization of Set Theory [1925]), quoted by Stewart Shapiro - Foundations without Foundationalism 8.2
Book Ref
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.209
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.
| 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] |
| 18179 | For Von Neumann the successor of n is n U {n} (rather than {n}) [Neumann, by Maddy] |
| 18180 | Von Neumann numbers are preferred, because they continue into the transfinite [Maddy on Neumann] |
| 15925 | Each Von Neumann ordinal number is the set of its predecessors [Neumann, by Lavine] |
| 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] |
| 22716 | Von Neumann defined ordinals as the set of all smaller ordinals [Neumann, by Poundstone] |
| 3355 | Von Neumann wanted mathematical functions to replace sets [Neumann, by Benardete,JA] |
| 3340 | Von Neumann defines each number as the set of all smaller numbers [Neumann, by Blackburn] |