72 ideas
15477 | Ontology is highly abstract physics, containing placeholders and exclusions [Martin,CB] |
15471 | Truth is a relation between a representation ('bearer') and part of the world ('truthmaker') [Martin,CB] |
15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
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] |
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] |
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] |
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] |
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] |
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] |
15484 | A property is a combination of a disposition and a quality [Martin,CB] |
15478 | Properties are the respects in which objects resemble, which places them in classes [Martin,CB] |
15483 | Properties are ways particular things are, and so they are tied to the identity of their possessor [Martin,CB] |
15480 | Objects are not bundles of tropes (which are ways things are, not parts of things) [Martin,CB] |
15489 | A property that cannot interact is worse than inert - it isn't there at all [Martin,CB] |
15487 | If unmanifested partnerless dispositions are still real, and are not just qualities, they can explain properties [Martin,CB] |
15479 | Properties endow a ball with qualities, and with powers or dispositions [Martin,CB] |
15488 | Qualities and dispositions are aspects of properties - what it exhibits, and what it does [Martin,CB] |
15469 | Dispositions in action can be destroyed, be recovered, or remain unchanged [Martin,CB] |
15467 | Powers depend on circumstances, so can't be given a conditional analysis [Martin,CB] |
15466 | 'The wire is live' can't be analysed as a conditional, because a wire can change its powers [Martin,CB] |
15476 | Structural properties involve dispositionality, so cannot be used to explain it [Martin,CB] |
15465 | Structures don't explain dispositions, because they consist of dispositions [Martin,CB] |
15481 | I favour the idea of a substratum for properties; spacetime seems to be just a bearer of properties [Martin,CB] |
15474 | Properly understood, wholes do no more causal work than their parts [Martin,CB] |
15486 | Only abstract things can have specific and full identity specifications [Martin,CB] |
15475 | The concept of 'identity' must allow for some changes in properties or parts [Martin,CB] |
15472 | It is pointless to say possible worlds are truthmakers, and then deny that possible worlds exist [Martin,CB] |
15492 | Explanations are mind-dependent, theory-laden, and interest-relative [Martin,CB] |
15495 | Analogy works, as when we eat food which others seem to be relishing [Martin,CB] |
15493 | Memory requires abstraction, as reminders of what cannot be fully remembered [Martin,CB] |
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] |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
15485 | Instead of a cause followed by an effect, we have dispositions in reciprocal manifestation [Martin,CB] |
15491 | Causation should be explained in terms of dispositions and manifestations [Martin,CB] |
15468 | Causal counterfactuals are just clumsy linguistic attempts to indicate dispositions [Martin,CB] |
15470 | Causal laws are summaries of powers [Martin,CB] |
10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
15482 | We can't think of space-time as empty and propertyless, and it seems to be a substratum [Martin,CB] |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |