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Single Idea 15524

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers ]

Full Idea

What sets Zermelo's modelling of arithmetic apart from von Neumann's and all the rest is that he identifies the primitive of arithmetic with an appropriately primitive notion of set theory.

Gist of Idea

Zermelo's model of arithmetic is distinctive because it rests on a primitive of set theory

Source

David Lewis (Parts of Classes [1991], 4.6)

Book Ref

Lewis,David: 'Parts of Classes' [Blackwell 1991], p.111

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A Reaction

Zermelo's model is just endlessly nested empty sets, which is a very simple structure. I gather that connoisseurs seem to prefer von Neumann's model (where each number contains its predecessor number).

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The 3 ideas with the same theme [Zermelo's view of numbers as nested sets]:

18178

For Zermelo the successor of n is {n} (rather than n U {n}) [Zermelo, by Maddy]

15524

Zermelo's model of arithmetic is distinctive because it rests on a primitive of set theory [Lewis]

8762	Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]