Ideas from 'Introduction to Zermelo's 1930 paper' by Michael Hallett [1996], by Theme Structure
[found in 'From Kant to Hilbert: sourcebook Vol. 2' (ed/tr Ewald,William) [OUP 1996,0198505361]].
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
17833

The firstorder ZF axiomatisation is highly noncategorical

17832

Zermelo showed that the ZF axioms in 1930 were noncategorical

17834

Noncategoricity reveals a sort of incompleteness, with sets existing that the axioms don't reveal

17835

Gödel show that the incompleteness of set theory was a necessity

4. Formal Logic / F. Set Theory ST / 7. Natural Sets
17837

Zermelo allows urelements, to enable the widespread application of settheory

6. Mathematics / A. Nature of Mathematics / 4. The Infinite / g. Continuum Hypothesis
17836

The General Continuum Hypothesis and its negation are both consistent with ZF
