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Single Idea 10883

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

Full Idea

Cantor's Continuum Hypothesis states that there are no sets which are too large for there to be a one-to-one correspondence between the set and the natural numbers, but too small for there to exist a one-to-one correspondence with the real numbers.

Gist of Idea

Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers

Source

report of George Cantor (works [1880]) by Leon Horsten - Philosophy of Mathematics §5.1

Book Ref

'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.22

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]