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Single Idea 9627

[filed under theme 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 Ref

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.

The 14 ideas from 'Investigations in the Foundations of Set Theory I'

13017 Zermelo introduced Pairing in 1930, and it seems fairly obvious [Zermelo, by Maddy]
13015 Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy]
13486 Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Zermelo, by Hart,WD]
13020 The Axiom of Separation requires set generation up to one step back from contradiction [Zermelo, by Maddy]
13487 In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Zermelo, by Hart,WD]
18178 For Zermelo the successor of n is {n} (rather than n U {n}) [Zermelo, by Maddy]
13027 Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy]
9627 Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR]
15924 Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Zermelo, by Lavine]
10870 ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Zermelo, by Clegg]
13012 Zermelo published his axioms in 1908, to secure a controversial proof [Zermelo, by Maddy]
17609 Set theory can be reduced to a few definitions and seven independent axioms [Zermelo]
17608 We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo]
17607 Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo]