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Single Idea 8762

[from 'Thinking About Mathematics' by Stewart Shapiro, in 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers ]

Full Idea

Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3.

Clarification

'Null' is the empty set, usually represented by Greek phi

Gist of Idea

Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3

Source

Stewart Shapiro (Thinking About Mathematics [2000], 10.2)

Book Reference

Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.265


A Reaction

See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them.

Related Idea

Idea 645 If two is part of three then numbers aren't Forms, because they would all be intermingled [Aristotle]