Combining Texts

Ideas for 'works', 'Representation in Music' and 'Understanding the Infinite'

expand these ideas     |    start again     |     choose another area for these texts

display all the ideas for this combination of texts

37 ideas

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
15907 Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine]
15910 Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
15905 Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine]
15942 Every rational number, unlike every natural number, is divisible by some other number [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
9983 Cantor took the ordinal numbers to be primary [Cantor, by Tait]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
17798 Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
9971 Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait]
9892 Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
14136 A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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]
15922 For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
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]
18250 Cauchy gave a necessary condition for the convergence of a sequence [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
15904 The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
15912 Counting results in well-ordering, and well-ordering makes counting possible [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
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]
15949 The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine]
15947 The infinite is extrapolation from the experience of indefinitely large size [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / c. Potential infinite
15940 The intuitionist endorses only the potential infinite [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
13464 Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
10112 The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman]
15909 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
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]
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]
8733 The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
15915 Ordinals are basic to Cantor's transfinite, to count the sets [Lavine]
15917 Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine]
18174 Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy]
15893 Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
18173 Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy]
15918 Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine]