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Full Idea
Cantor's conjecture (the Continuum Hypothesis) is that there are no sets between N and P(N). The 'generalized' version replaces N with an arbitrary infinite set.
Clarification
'N' is the natural numbers, and 'P(N)' is their power set
Gist of Idea
Continuum Hypothesis: there are no sets between N and P(N)
Source
report of George Cantor (works [1880]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2
Book Ref
Wolf,Robert S.: 'A Tour Through Mathematical Logic' [Carus Maths Monographs 2005], p.67
A Reaction
The initial impression is that there is a single gap in the numbers, like a hole in ozone layer, but the generalised version implies an infinity of gaps. How can there be gaps in the numbers? Weird.
10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
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 | 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 | 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] |