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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 49 ideas from George Cantor

9145 We form the image of a cardinal number by a double abstraction, from the elements and from their order [Cantor]
15911 Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine]
15946 Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine]
9992 The 'extension of a concept' in general may be quantitatively completely indeterminate [Cantor]
17831 Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Cantor, by Lake]
15896 Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine]
9616 A set is a collection into a whole of distinct objects of our intuition or thought [Cantor]
13444 Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD]
18098 Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock]
10865 The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg]
15505 If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis]
10701 Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter]
10232 Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro]
8631 Cantor says that maths originates only by abstraction from objects [Cantor, by Frege]
8715 Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend]
13454 Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor]
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]
18174 Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy]
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]
18173 Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy]
13016 The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy]
14199 Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley]
10082 There are infinite sets that are not enumerable [Cantor, by Smith,P]
13483 Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD]
8710 The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend]
15910 Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine]
15905 Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine]
9983 Cantor took the ordinal numbers to be primary [Cantor, by Tait]
17798 Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry]
9971 Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait]
15893 Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine]
9892 Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett]
14136 A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor]
15906 Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine]
11015 Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read]
15903 A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine]
18251 Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine]
15902 Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine]
15908 It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine]
13464 Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD]
10112 The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman]
8733 The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro]
13465 Only God is absolutely infinite [Cantor, by Hart,WD]
10863 Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg]
15901 Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine]
18176 Pure mathematics is pure set theory [Cantor]