Combining Texts

Ideas for 'Necessary Existents', 'On the Introduction of Transfinite Numbers' and 'Word and Object'

unexpand these ideas     |    start again     |     choose another area for these texts

display all the ideas for this combination of texts

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
17905 Any progression will do nicely for numbers; they can all then be used to measure multiplicity [Quine]
     Full Idea: The condition on an explication of number can be put succinctly: any progression will do nicely. Russell once held that one must also be able to measure multiplicity, but this was a mistake; any progression can be fitted to that further condition.
     From: Willard Quine (Word and Object [1960], §54)
     A reaction: [compressed] This is the strongest possible statement that the numbers are the ordinals, and the Peano Axioms will define them. The Fregean view that cardinality comes first is redundant.
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