Ideas from 'Investigations in the Foundations of Set Theory I' by Ernst Zermelo [1908], by Theme Structure
[found in 'From Frege to Gödel 18791931' (ed/tr Heijenoort,Jean van) [Harvard 1967,0674324498]].
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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?

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
13012

Zermelo published his axioms in 1908, to secure a controversial proof

17609

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
13017

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
13015

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
13020

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
13487

In ZF, the BuraliForti 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
13027

Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets
