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'reports', 'works' and 'Intro to 'The Reason's Proper Study''
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31 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
15910
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Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
15905
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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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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
9983
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Cantor took the ordinal numbers to be primary [Cantor, by Tait]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
17798
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Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
9971
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Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait]
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9892
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Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
14136
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A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
15906
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Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine]
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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 [Cantor, by Read]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
15903
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A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine]
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18251
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Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
15902
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Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine]
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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 [Cantor, by Lavine]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
13464
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Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
10112
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The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
8733
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The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro]
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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 [Cantor, by Koellner]
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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 [Cantor, by Hart,WD]
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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 [Cantor, by Horsten]
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13528
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Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS]
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9555
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Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
18174
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Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy]
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15893
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Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
18173
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Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
10624
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The incompletability of formal arithmetic reveals that logic also cannot be completely characterized [Hale/Wright]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
10232
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Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
18176
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Pure mathematics is pure set theory [Cantor]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
10629
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If structures are relative, this undermines truth-value and objectivity [Hale/Wright]
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10628
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The structural view of numbers doesn't fit their usage outside arithmetical contexts [Hale/Wright]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
8631
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Cantor says that maths originates only by abstraction from objects [Cantor, by Frege]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
10622
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The neo-Fregean is more optimistic than Frege about contextual definitions of numbers [Hale/Wright]
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