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Single Idea 8733
[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
]
Full Idea
Cantor's 'continuum hypothesis' is the assertion that there are no infinite cardinalities strictly between the size of the natural numbers and the size of the real numbers.
Gist of Idea
The Continuum Hypothesis says there are no sets between the natural numbers and reals
Source
report of George Cantor (works [1880]) by Stewart Shapiro - Thinking About Mathematics 2.4
Book Ref
Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.42
A Reaction
The tricky question is whether this hypothesis can be proved.
The
15 ideas
with the same theme
[denial of a cardinality between naturals are reals]:
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8733
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The Continuum Hypothesis says there are no sets between the natural numbers and reals
[Cantor, by Shapiro]
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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]
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13447
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Cantor: there is no size between naturals and reals, or between a set and its power set
[Cantor, by Hart,WD]
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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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13528
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Continuum Hypothesis: there are no sets between N and P(N)
[Cantor, by Wolf,RS]
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9555
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Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum
[Cantor, by Chihara]
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10868
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The Continuum Hypothesis is not inconsistent with the axioms of set theory
[Gödel, by Clegg]
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13517
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If set theory is consistent, we cannot refute or prove the Continuum Hypothesis
[Gödel, by Hart,WD]
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10046
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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]
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17836
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The General Continuum Hypothesis and its negation are both consistent with ZF
[Hallett,M]
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17615
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Every infinite set of reals is either countable or of the same size as the full set of reals
[Maddy]
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13652
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The 'continuum' is the cardinality of the powerset of a denumerably infinite set
[Shapiro]
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10862
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The 'continuum hypothesis' says aleph-one is the cardinality of the reals
[Clegg]
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10869
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The Continuum Hypothesis is independent of the axioms of set theory
[Clegg]
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