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Single Idea 13012

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets ]

Full Idea

Zermelo proposed his listed of assumptions (including the controversial Axiom of Choice) in 1908, in order to secure his controversial proof of Cantor's claim that ' we can always bring any well-defined set into the form of a well-ordered set'.

Gist of Idea

Zermelo published his axioms in 1908, to secure a controversial proof

Source

report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1

Book Ref

-: 'Journal of Symbolic Logic' [-], p.483

A Reaction

This is interesting because it sometimes looks as if axiom systems are just a way of tidying things up. Presumably it is essential to get people to accept the axioms in their own right, the 'old-fashioned' approach that they be self-evident.

The 21 ideas from Ernst Zermelo

17832 Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M]
13028 Replacement was added when some advanced theorems seemed to need it [Zermelo, by Maddy]
17626 The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo]
13487 In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Zermelo, by Hart,WD]
13015 Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy]
13020 The Axiom of Separation requires set generation up to one step back from contradiction [Zermelo, by Maddy]
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]
13486 Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Zermelo, by Hart,WD]
13017 Zermelo introduced Pairing in 1930, and it seems fairly obvious [Zermelo, by Maddy]
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]
17613 We should judge principles by the science, not science by some fixed principles [Zermelo]
15897 Zermelo realised that Choice would facilitate the sort of 'counting' Cantor needed [Zermelo, by Lavine]
9565 Zermelo made 'set' and 'member' undefined axioms [Zermelo, by Chihara]
3339 For Zermelo's set theory the empty set is zero and the successor of each number is its unit set [Zermelo, by Blackburn]