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Ideas of Ernst Zermelo, by Text

[German, 1871 - 1953, Professor at Göttingen, and then at Zurich.]

1904 Proof that every set can be well-ordered
p.4 Zermelo realised that Choice would facilitate the sort of 'counting' Cantor needed [Lavine]
1908 Investigations in the Foundations of Set Theory I
p.24 For Zermelo the successor of n is {n} (rather than n U {n}) [Maddy]
p.60 Different versions of set theory result in different underlying structures for numbers [Brown,JR]
p.69 Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Hart,WD]
p.70 In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Hart,WD]
p.107 Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Lavine]
p.205 ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Clegg]
p.483 Zermelo published his axioms in 1908, to secure a controversial proof [Maddy]
p.484 Zermelo introduced Pairing in 1930, and it seems fairly obvious [Maddy]
p.484 Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Maddy]
p.485 The Axiom of Separation requires set generation up to one step back from contradiction [Maddy]
p.489 Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Maddy]
Intro p.200 Set theory can be reduced to a few definitions and seven independent axioms
Intro p.200 Set theory investigates number, order and function, showing logical foundations for mathematics
Intro p.200 We take set theory as given, and retain everything valuable, while avoiding contradictions
1908 New Proof of Possibility of Well-Ordering
§2a p.189 We should judge principles by the science, not science by some fixed principles
1920 works
p.204 Zermelo made 'set' and 'member' undefined axioms [Chihara]
p.280 For Zermelo's set theory the empty set is zero and the successor of each number is its unit set [Blackburn]
1930 On boundary numbers and domains of sets
p.489 Replacement was added when some advanced theorems seemed to need it [Maddy]
p.1209 Zermelo showed that the ZF axioms in 1930 were non-categorical [Hallett,M]
§5 p.1233 The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers