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Single Idea 13517

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

Full Idea

Gödel proved that (if set theory is consistent) we cannot refute the continuum hypothesis, and Cohen proved that (if set theory is consistent) we cannot prove it either.

Clarification

The hypothesis is that the numbers contain gaps

Gist of Idea

If set theory is consistent, we cannot refute or prove the Continuum Hypothesis

Source

report of Kurt Gödel (What is Cantor's Continuum Problem? [1964]) by William D. Hart - The Evolution of Logic 10

Book Ref

Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.273

The 15 ideas with the same theme [denial of a cardinality between naturals are reals]:

8733 The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro]
17889 CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner]
13447 Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD]
10883 Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten]
13528 Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS]
9555 Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara]
10868 The Continuum Hypothesis is not inconsistent with the axioms of set theory [Gödel, by Clegg]
13517 If set theory is consistent, we cannot refute or prove the Continuum Hypothesis [Gödel, by Hart,WD]
10046 The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers [Gödel]
12327 The undecidability of the Continuum Hypothesis may have ruined or fragmented set theory [Badiou]
17836 The General Continuum Hypothesis and its negation are both consistent with ZF [Hallett,M]
17615 Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy]
13652 The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro]
10862 The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg]
10869 The Continuum Hypothesis is independent of the axioms of set theory [Clegg]