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Single Idea 17889
[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
]
Full Idea
Cantor's Continuum Hypothesis (CH) says that for every infinite set X of reals there is either a one-to-one correspondence between X and the natural numbers, or between X and the real numbers.
Gist of Idea
CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals
Source
report of George Cantor (works [1880]) by Peter Koellner - On the Question of Absolute Undecidability 1.2
Book Ref
-: 'Philosophia Mathematica' [-], p.7
A Reaction
Every single writer I read defines this differently, which drives me crazy, but is also helpfully illuminating. There is a moral there somewhere.
The
15 ideas
with the same theme
[denial of a cardinality between naturals are reals]:
10883
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Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers
[Cantor, by Horsten]
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8733
|
The Continuum Hypothesis says there are no sets between the natural numbers and reals
[Cantor, by Shapiro]
|
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]
|
17889
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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]
|
10868
|
The Continuum Hypothesis is not inconsistent with the axioms of set theory
[Gödel, by Clegg]
|
13517
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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]
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12327
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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]
|