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?
4. Formal Logic / F. Set Theory ST / 1. Set Theory
We take set theory as given, and retain everything valuable, while avoiding contradictions
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
For Zermelo's set theory the empty set is zero and the successor of each number is its unit set
Zermelo made 'set' and 'member' undefined axioms
Zermelo published his axioms in 1908, to secure a controversial proof
Set theory can be reduced to a few definitions and seven independent axioms
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
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
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / m. Axiom of Separation
The Axiom of Separation requires set generation up to one step back from contradiction
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. Numbers / e. Ordinal numbers
In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / e. Countable infinity
Zermelo realised that Choice would facilitate the sort of 'counting' Cantor needed
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets