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
The first-order ZF axiomatisation is highly non-categorical
Zermelo showed that the ZF axioms in 1930 were non-categorical
Non-categoricity reveals a sort of incompleteness, with sets existing that the axioms don't reveal
Gödel show that the incompleteness of set theory was a necessity
4. Formal Logic / F. Set Theory ST / 7. Natural Sets
Zermelo allows ur-elements, to enable the widespread application of set-theory
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / g. Continuum Hypothesis
The General Continuum Hypothesis and its negation are both consistent with ZF