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Single Idea 15893

[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity ]

Full Idea

Cantor's set theory was not of collections in some familiar sense, but of collections that can be counted using the indexes - the finite and transfinite ordinal numbers. ..He treated infinite collections as if they were finite.

Gist of Idea

Cantor's theory concerns collections which can be counted, using the ordinals

Source

report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I

Book Ref

Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.3

Related Idea

Idea 15896 Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine]


The 42 ideas from 'works'

Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine]
Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD]
Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock]
If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis]
The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg]
Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter]
The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy]
Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley]
There are infinite sets that are not enumerable [Cantor, by Smith,P]
Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD]
The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend]
Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine]
Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy]
Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine]
Cantor took the ordinal numbers to be primary [Cantor, by Tait]
Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry]
Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait]
Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett]
A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor]
Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine]
Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read]
A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine]
Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine]
Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine]
It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine]
Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD]
Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine]
The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman]
CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner]
The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro]
Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD]
Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten]
Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS]
Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara]
Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy]
Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro]
Cantor says that maths originates only by abstraction from objects [Cantor, by Frege]
Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend]
Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor]
Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg]
Only God is absolutely infinite [Cantor, by Hart,WD]
Pure mathematics is pure set theory [Cantor]