97 ideas
| 24261 | Devotion to learning and applied intelligence leads to divine wisdom - if truth is available [Plato] |
| 326 | For relaxation one can consider the world of change, instead of eternal things [Plato] |
| 315 | Philosophy is the supreme gift of the gods to mortals [Plato] |
| 306 | Nothing can come to be without a cause [Plato] |
| 24250 | We should not pick out 'this' water, but only 'something of this sort' [Plato] |
| 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] |
| 8920 | Equivalence relations are reflexive, symmetric and transitive, and classify similar objects [Lipschutz] |
| 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] |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| 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] |
| 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] |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| 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] |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| 24246 | The sun was made for light, so we could learn numbers from astronomical movement [Plato] |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| 324 | Before the existence of the world there must have been being, space and becoming [Plato] |
| 20364 | The apprehensions of reason remain unchanging, but reasonless sensation shows mere becoming [Plato] |
| 321 | For knowledge and true opinion to be different there must be Forms; otherwise we are just stuck with sensations [Plato] |
| 307 | Something will always be well-made if the maker keeps in mind the eternal underlying pattern [Plato] |
| 318 | In addition to the underlying unchanging model and a changing copy of it, there must also be a foundation of all change [Plato] |
| 12042 | Plato's Forms were seen as part of physics, rather than of metaphysics [Plato, by Annas] |
| 317 | The universe is basically an intelligible and unchanging model, and a visible and changing copy of it [Plato] |
| 24254 | Two existing entities can never strictly coincide [Plato] |
| 24236 | Some statements about what is obvious and stable are as irrefutable as possible [Plato] |
| 24252 | Knowledge is taught, has logos, is unshakeable, and is rare [Plato] |
| 24251 | If knowledge is just true belief, we are forced to rely on the senses [Plato] |
| 334 | Only bird-brained people think astronomy is entirely a matter of evidence [Plato] |
| 5962 | Plato says the soul is ordered by number [Plato, by Plutarch] |
| 24244 | The soul is a complex mixture of pure mind and changing matter [Plato] |
| 24258 | The gods placed the mortal soul in the chest [Plato] |
| 330 | No one wants to be bad, but bad men result from physical and educational failures, which they do not want or choose [Plato] |
| 24240 | Intelligence requires soul [Plato] |
| 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] |
| 24241 | Beauty must always be perfect [Plato] |
| 316 | Music has harmony like the soul, and serves to reorder disharmony within us [Plato] |
| 24260 | The best part of the soul raises us up to the heavens, to which we are naturally akin [Plato] |
| 24259 | Death in old age is a natural end, untroubled, and more pleasure than distress [Plato] |
| 24239 | Perfect goodness always produces perfect beauty [Plato] |
| 332 | One should exercise both the mind and the body, to avoid imbalance [Plato] |
| 24257 | Unnatural modifications are painful, and restoring normality is pleasant [Plato] |
| 328 | Everything that takes place naturally is pleasant [Plato] |
| 24234 | I have discussed the best constitution, and the kind of citizens it requires [Plato] |
| 24235 | Female Guardians will have identical duties to the men [Plato] |
| 24248 | The god said human nature comes as the superior male, and inferior female [Plato] |
| 322 | Intelligence is the result of rational teaching; true opinion can result from irrational persuasion [Plato] |
| 331 | Bad governments prevent discussion, and discourage the study of virtue [Plato] |
| 310 | The creator of the cosmos had no envy, and so wanted things to be as like himself as possible [Plato] |
| 311 | The cosmos must be unique, because it resembles the creator, who is unique [Plato] |
| 24249 | The elements seem able to transmute into each other [Plato] |
| 24243 | The world-maker used the four elements and their properties in entirety [Plato] |
| 325 | We must consider the four basic shapes as too small to see, only becoming visible in large numbers [Plato] |
| 327 | There are two types of cause, the necessary and the divine [Plato] |
| 24255 | Motion needs differing moved and mover, so it originates in diversity [Plato] |
| 24256 | The spherical universe composed of four elements squeezes out every bit of void [Plato] |
| 24253 | Space is eternal and indestructible, but is only known by barely credible reasoning [Plato] |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| 24245 | The god created eternity in the sequence of the universe, and its image we call 'time' [Plato] |
| 314 | Heavenly movements gave us the idea of time, and caused us to inquire about the heavens [Plato] |
| 312 | Time came into existence with the heavens, so that there will be a time when they can be dissolved [Plato] |
| 309 | Clearly the world is good, so its maker must have been concerned with the eternal, not with change [Plato] |
| 308 | If the cosmos is an object of perception then it must be continually changing [Plato] |
| 24238 | The god found chaos, and led it to superior order [Plato] |
| 24242 | Is there a plurality (or even an infinity) of universes? No, because the model makes it unique [Plato] |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| 24247 | The universe has four types of living being: gods, birds, fish, and land animals [Plato] |
| 24237 | The divine organiser of the world wanted it to have as little imperfection as possible [Plato] |