70 ideas
8623 | Proof reveals the interdependence of truths, as well as showing their certainty [Euclid, by Frege] |
13907 | If you pick an arbitrary triangle, things proved of it are true of all triangles [Euclid, by Lemmon] |
15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
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] |
8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
6297 | Euclid's geometry is synthetic, but Descartes produced an analytic version of it [Euclid, by Resnik] |
15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
9603 | An assumption that there is a largest prime leads to a contradiction [Euclid, by Brown,JR] |
9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
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] |
14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
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] |
9894 | A unit is that according to which each existing thing is said to be one [Euclid] |
8738 | Postulate 2 says a line can be extended continuously [Euclid, by Shapiro] |
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] |
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] |
17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
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] |
13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
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] |
18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
10302 | Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Euclid, by Bernays] |
22278 | Euclid relied on obvious properties in diagrams, as well as on his axioms [Potter on Euclid] |
8673 | Euclid's parallel postulate defines unique non-intersecting parallel lines [Euclid, by Friend] |
10250 | Euclid needs a principle of continuity, saying some lines must intersect [Shapiro on Euclid] |
14157 | Modern geometries only accept various parts of the Euclid propositions [Russell on Euclid] |
1600 | Euclid's common notions or axioms are what we must have if we are to learn anything at all [Euclid, by Roochnik] |
10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
18176 | Pure mathematics is pure set theory [Cantor] |
8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
3847 | Man is nothing else but the sum of his actions [Sartre] |
3846 | Man IS freedom [Sartre] |
8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
3843 | There is no human nature [Sartre] |
20762 | There are no values to justify us, and no excuses [Sartre] |
3852 | If values depend on us, freedom is the foundation of all values [Sartre] |
20764 | In becoming what we want to be we create what we think man ought to be [Sartre] |
3848 | Cowards are responsible for their cowardice [Sartre] |
20763 | When my personal freedom becomes involved, I must want freedom for everyone else [Sartre] |
22229 | Existentialists says that cowards and heroes make themselves [Sartre] |
3842 | Existence before essence (or begin with the subjective) [Sartre] |
6868 | 'Existence precedes essence' means we have no pre-existing self, but create it through existence [Sartre, by Le Poidevin] |
3844 | Existentialism says man is whatever he makes of himself [Sartre] |
20754 | It is dishonest to offer passions as an excuse [Sartre] |
6571 | When a man must choose between his mother and the Resistance, no theory can help [Sartre, by Fogelin] |
3851 | If I do not choose, that is still a choice [Sartre] |
10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
3845 | Without God there is no intelligibility or value [Sartre] |