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
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
Set theory investigates number, order and function, showing logical foundations for mathematics
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
ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Clegg]
Zermelo published his axioms in 1908, to secure a controversial proof [Maddy]
Set theory can be reduced to a few definitions and seven independent axioms
Zermelo made 'set' and 'member' undefined axioms [Chihara]
For Zermelo's set theory the empty set is zero and the successor of each number is its unit set [Blackburn]
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
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
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
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
Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Hart,WD]
The Axiom of Separation requires set generation up to one step back from contradiction [Maddy]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
We should judge principles by the science, not science by some fixed principles
5. Theory of Logic / L. Paradox / 3. Antinomies
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
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
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
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
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
Different versions of set theory result in different underlying structures for numbers [Brown,JR]