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Single Idea 17836

[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis ]

Full Idea

In 1938, Gödel showed that ZF plus the General Continuum Hypothesis is consistent if ZF is. Cohen showed that ZF and not-GCH is also consistent if ZF is, which finally shows that neither GCH nor ¬GCH can be proved from ZF itself.

Gist of Idea

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

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]
17837 Zermelo allows ur-elements, to enable the widespread application of set-theory [Hallett,M]
17836 The General Continuum Hypothesis and its negation are both consistent with ZF [Hallett,M]