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Single Idea 9627

[from 'Investigations in the Foundations of Set Theory I' by Ernst Zermelo, in 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique ]

Full Idea

In Zermelo's set-theoretic definition of number, 2 is a member of 3, but not a member of 4; in Von Neumann's definition every number is a member of every larger number. This means they have two different structures.

Gist of Idea

Different versions of set theory result in different underlying structures for numbers

Source

report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by James Robert Brown - Philosophy of Mathematics Ch. 4

Book Reference

Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.60


A Reaction

This refers back to the dilemma highlighted by Benacerraf, which was supposed to be the motivation for structuralism. My intuition says that the best answer is that they are both wrong. In a pattern, the nodes aren't 'members' of one another.