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Single Idea 17837

[filed under theme 4. Formal Logic / F. Set Theory ST / 7. Natural Sets ]

Full Idea

Unlike earlier writers (such as Fraenkel), Zermelo clearly allows that there might be ur-elements (that is, objects other than the empty set, which have no members). Indeed he sees in this the possibility of widespread application of set-theory.

Gist of Idea

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

Source

Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1217)

Book Ref

'From Kant to Hilbert: sourcebook Vol. 2', ed/tr. Ewald,William [OUP 1996], p.1217

The 4 ideas from 'Introduction to Zermelo's 1930 paper'

17833 The first-order ZF axiomatisation is highly non-categorical [Hallett,M]
17834 Non-categoricity reveals a sort of incompleteness, with sets existing that the axioms don't reveal [Hallett,M]
17836 The General Continuum Hypothesis and its negation are both consistent with ZF [Hallett,M]
17837 Zermelo allows ur-elements, to enable the widespread application of set-theory [Hallett,M]