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Single Idea 10082
[filed under theme 5. Theory of Logic / K. Features of Logics / 8. Enumerability
]
Full Idea
Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'.
Gist of Idea
There are infinite sets that are not enumerable
Source
report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.14
A Reaction
[Smith summarises the diagonal argument]
The
42 ideas
from 'works'
15901
|
Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory
[Cantor, by Lavine]
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13444
|
Cantor's Theorem: for any set x, its power set P(x) has more members than x
[Cantor, by Hart,WD]
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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]
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15505
|
If a set is 'a many thought of as one', beginners should protest against singleton sets
[Cantor, by Lewis]
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10701
|
Cantor showed that supposed contradictions in infinity were just a lack of clarity
[Cantor, by Potter]
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13016
|
The Axiom of Union dates from 1899, and seems fairly obvious
[Cantor, by Maddy]
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14199
|
Cantor's sets were just collections, but Dedekind's were containers
[Cantor, by Oliver/Smiley]
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10082
|
There are infinite sets that are not enumerable
[Cantor, by Smith,P]
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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]
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8710
|
The powerset of all the cardinal numbers is required to be greater than itself
[Cantor, by Friend]
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15910
|
Cantor named the third realm between the finite and the Absolute the 'transfinite'
[Cantor, by Lavine]
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15905
|
Cantor proved the points on a plane are in one-to-one correspondence to the points on a line
[Cantor, by Lavine]
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9983
|
Cantor took the ordinal numbers to be primary
[Cantor, by Tait]
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17798
|
Cantor presented the totality of natural numbers as finite, not infinite
[Cantor, by Mayberry]
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9971
|
Cantor introduced the distinction between cardinals and ordinals
[Cantor, by Tait]
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15893
|
Cantor's theory concerns collections which can be counted, using the ordinals
[Cantor, by Lavine]
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9892
|
Cantor showed that ordinals are more basic than cardinals
[Cantor, by Dummett]
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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]
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11015
|
Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1
[Cantor, by Read]
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15903
|
A real is associated with an infinite set of infinite Cauchy sequences of rationals
[Cantor, by Lavine]
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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]
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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]
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8733
|
The Continuum Hypothesis says there are no sets between the natural numbers and reals
[Cantor, by Shapiro]
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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]
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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]
|
10232
|
Property extensions outstrip objects, so shortage of objects caused the Caesar problem
[Cantor, by Shapiro]
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8631
|
Cantor says that maths originates only by abstraction from objects
[Cantor, by Frege]
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8715
|
Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability
[Cantor, by Friend]
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13454
|
Cantor says (vaguely) that we abstract numbers from equal sized sets
[Hart,WD on Cantor]
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13465
|
Only God is absolutely infinite
[Cantor, by Hart,WD]
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10863
|
Cantor proved that three dimensions have the same number of points as one dimension
[Cantor, by Clegg]
|
18176
|
Pure mathematics is pure set theory
[Cantor]
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