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 1996,0-19-850536-1]].

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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets

17833

The first-order ZF axiomatisation is highly non-categorical

17834

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

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

17837

Zermelo allows ur-elements, to enable the widespread application of set-theory

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis

17836

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