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Ideas of George Cantor, by Text
[German, 1845 - 1918, Born in St Petersburg. Studied in Berlin. Taught at the University of Halle from 1872.]
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p.3
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10701
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Cantor showed that supposed contradictions in infinity were just a lack of clarity [Potter]
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p.3
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15893
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Cantor's theory concerns collections which can be counted, using the ordinals [Lavine]
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p.7
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15901
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Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Lavine]
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p.7
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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 [Koellner]
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p.11
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15902
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Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Lavine]
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p.14
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10082
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There are infinite sets that are not enumerable [Smith,P]
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p.16
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13444
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Cantor's Theorem: for any set x, its power set P(x) has more members than x [Hart,WD]
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p.17
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18174
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Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Maddy]
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p.17
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18173
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Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Maddy]
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p.19
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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 [Hart,WD]
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p.19
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13454
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Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD]
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p.22
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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 [Horsten]
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p.27
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8631
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Cantor says that maths originates only by abstraction from objects [Frege]
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p.29
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13465
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Only God is absolutely infinite [Hart,WD]
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p.29
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13464
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Cantor proposes that there won't be a potential infinity if there is no actual infinity [Hart,WD]
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p.30
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15505
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If a set is 'a many thought of as one', beginners should protest against singleton sets [Lewis]
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p.38
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15903
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A real is associated with an infinite set of infinite Cauchy sequences of rationals [Lavine]
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p.40
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9971
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Cantor introduced the distinction between cardinals and ordinals [Tait]
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p.42
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8733
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The Continuum Hypothesis says there are no sets between the natural numbers and reals [Shapiro]
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p.42
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15905
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Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Lavine]
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p.43
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15906
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Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Lavine]
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p.48
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15908
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It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Lavine]
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p.49
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9983
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Cantor took the ordinal numbers to be primary [Tait]
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p.51
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15910
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Cantor named the third realm between the finite and the Absolute the 'transfinite' [Lavine]
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p.60
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13483
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Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Hart,WD]
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p.67
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13528
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Continuum Hypothesis: there are no sets between N and P(N) [Wolf,RS]
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p.79
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9555
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Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Chihara]
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p.92
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18251
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Irrational numbers are the limits of Cauchy sequences of rational numbers [Lavine]
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p.99
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10112
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The naturals won't map onto the reals, so there are different sizes of infinity [George/Velleman]
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p.106
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18098
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Cantor proved that all sets have more subsets than they have members [Bostock]
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p.113
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8710
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The powerset of all the cardinal numbers is required to be greater than itself [Friend]
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p.123
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14199
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Cantor's sets were just collections, but Dedekind's were containers [Oliver/Smiley]
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p.127
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10232
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Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Shapiro]
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p.145
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8715
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Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Friend]
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p.163
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11015
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Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Read]
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p.183
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10863
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Cantor proved that three dimensions have the same number of points as one dimension [Clegg]
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p.185
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10865
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The continuum is the powerset of the integers, which moves up a level [Clegg]
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p.293
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9892
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Cantor showed that ordinals are more basic than cardinals [Dummett]
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p.304
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14136
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A cardinal is an abstraction, from the nature of a set's elements, and from their order
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p.414
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17798
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Cantor presented the totality of natural numbers as finite, not infinite [Mayberry]
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p.484
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13016
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The Axiom of Union dates from 1899, and seems fairly obvious [Maddy]
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I.1
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p.21
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18176
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Pure mathematics is pure set theory
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1883
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Grundlagen (Foundations of Theory of Manifolds)
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p.52
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15911
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Ordinals are generated by endless succession, followed by a limit ordinal [Lavine]
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p.247
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15946
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Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Lavine]
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1885
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Review of Frege's 'Grundlagen'
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1932:440
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p.60
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9992
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The 'extension of a concept' in general may be quantitatively completely indeterminate
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1897
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The Theory of Transfinite Numbers
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p.4
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15896
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Cantor needed Power Set for the reals, but then couldn't count the new collections [Lavine]
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p.85
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p.22
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9616
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A set is a collection into a whole of distinct objects of our intuition or thought
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1899
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Later Letters to Dedekind
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p.366
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17831
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Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Lake]
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§1
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p.599
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9145
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We form the image of a cardinal number by a double abstraction, from the elements and from their order
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