Ideas from 'Investigations in the Foundations of Set Theory I' by Ernst Zermelo [1908], by Theme Structure

[found in 'From Frege to Gödel 1879-1931' (ed/tr Heijenoort,Jean van) [Harvard 1967,0-674-32449-8]].

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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
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
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 / 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