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'works', 'On the Introduction of Transfinite Numbers' and 'Comment on Armstrong and Forrest'
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5 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
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13489
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Von Neumann treated cardinals as a special sort of ordinal [Neumann, by Hart,WD]
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Full Idea:
Von Neumann's decision was to start with the ordinals and to treat cardinals as a special sort of ordinal.
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From:
report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by William D. Hart - The Evolution of Logic 3
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A reaction:
[see Hart 73-74 for an explication of this]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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12336
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A von Neumann ordinal is a transitive set with transitive elements [Neumann, by Badiou]
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Full Idea:
In Von Neumann's definition an ordinal is a transitive set in which all of the elements are transitive.
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From:
report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Alain Badiou - Briefings on Existence 11
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / g. Von Neumann numbers
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18179
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For Von Neumann the successor of n is n U {n} (rather than {n}) [Neumann, by Maddy]
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Full Idea:
For Von Neumann the successor of n is n U {n} (rather than Zermelo's successor, which is {n}).
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From:
report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n8
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18180
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Von Neumann numbers are preferred, because they continue into the transfinite [Maddy on Neumann]
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Full Idea:
Von Neumann's version of the natural numbers is in fact preferred because it carries over directly to the transfinite ordinals.
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From:
comment on John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n9
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15925
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Each Von Neumann ordinal number is the set of its predecessors [Neumann, by Lavine]
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Full Idea:
Each Von Neumann ordinal number is the set of its predecessors. ...He had shown how to introduce ordinal numbers as sets, making it possible to use them without leaving the domain of sets.
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From:
report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Shaughan Lavine - Understanding the Infinite V.3
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