### Ideas of Ernst Zermelo, by Theme

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

green numbers give full details    |    back to list of philosophers    |     expand these ideas
###### 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
 17608 We take set theory as given, and retain everything valuable, while avoiding contradictions
 17607 Set theory investigates number, order and function, showing logical foundations for mathematics
###### 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
 3339 For Zermelo's set theory the empty set is zero and the successor of each number is its unit set [Blackburn]
 9565 Zermelo made 'set' and 'member' undefined axioms [Chihara]
 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 / 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
 13486 Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Hart,WD]
 13020 The Axiom of Separation requires set generation up to one step back from contradiction [Maddy]
###### 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