Ideas of Ernst Zermelo, by Theme

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

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2. Reason / D. Definition / 8. Impredicative Definition
15924 Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Lavine]
4. Formal Logic / F. Set Theory ST / 1. Set Theory
17607 Set theory investigates number, order and function, showing logical foundations for mathematics
17608 We take set theory as given, and retain everything valuable, while avoiding contradictions
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
10870 ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Clegg]
13012 Zermelo published his axioms in 1908, to secure a controversial proof [Maddy]
17609 Set theory can be reduced to a few definitions and seven independent axioms
9565 Zermelo made 'set' and 'member' undefined axioms [Chihara]
3339 For Zermelo's set theory the empty set is zero and the successor of each number is its unit set [Blackburn]
17832 Zermelo showed that the ZF axioms in 1930 were non-categorical [Hallett,M]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
13017 Zermelo introduced Pairing in 1930, and it seems fairly obvious [Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
13028 Replacement was added when some advanced theorems seemed to need it [Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
13015 Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / m. Axiom of Separation
13020 The Axiom of Separation requires set generation up to one step back from contradiction [Maddy]
13486 Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Hart,WD]
5. Theory of Logic / L. Paradox / 3. Antinomies
17626 The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13487 In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Hart,WD]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / e. Countable infinity
15897 Zermelo realised that Choice would facilitate the sort of 'counting' Cantor needed [Lavine]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
18178 For Zermelo the successor of n is {n} (rather than n U {n}) [Maddy]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
13027 Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Maddy]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
9627 Different versions of set theory result in different underlying structures for numbers [Brown,JR]
18. Thought / A. Modes of Thought / 6. Judgement / a. Nature of Judgement
17613 We should judge principles by the science, not science by some fixed principles