34 ideas
| 1597 | Thales was the first western thinker to believe the arché was intelligible [Roochnik on Thales] |
| 10859 | A set is 'well-ordered' if every subset has a first element [Clegg] |
| 10857 | Set theory made a closer study of infinity possible [Clegg] |
| 10864 | Any set can always generate a larger set - its powerset, of subsets [Clegg] |
| 10872 | Extensionality: Two sets are equal if and only if they have the same elements [Clegg] |
| 10875 | Pairing: For any two sets there exists a set to which they both belong [Clegg] |
| 10876 | Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg] |
| 10878 | Infinity: There exists a set of the empty set and the successor of each element [Clegg] |
| 10877 | Powers: All the subsets of a given set form their own new powerset [Clegg] |
| 10879 | Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg] |
| 10871 | Axiom of Existence: there exists at least one set [Clegg] |
| 10874 | Specification: a condition applied to a set will always produce a new set [Clegg] |
| 10880 | Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg] |
| 10861 | Beyond infinity cardinals and ordinals can come apart [Clegg] |
| 10860 | An ordinal number is defined by the set that comes before it [Clegg] |
| 10854 | Transcendental numbers can't be fitted to finite equations [Clegg] |
| 10858 | By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg] |
| 10853 | Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg] |
| 10866 | Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg] |
| 10869 | The Continuum Hypothesis is independent of the axioms of set theory [Clegg] |
| 10862 | The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg] |
| 21110 | An understanding of the most basic physics should explain all of the subject's mysteries [Krauss] |
| 21105 | In 1676 it was discovered that water is teeming with life [Krauss] |
| 3013 | Nothing is stronger than necessity, which rules everything [Thales, by Diog. Laertius] |
| 1494 | Thales said water is the first principle, perhaps from observing that food is moist [Thales, by Aristotle] |
| 1713 | Thales must have thought soul causes movement, since he thought magnets have soul [Thales, by Aristotle] |
| 21109 | Space itself can expand (and separate its contents) at faster than light speeds [Krauss] |
| 21104 | General Relativity: the density of energy and matter determines curvature and gravity [Krauss] |
| 21107 | Uncertainty says that energy can be very high over very short time periods [Krauss] |
| 21106 | Most of the mass of a proton is the energy in virtual particles (rather than the quarks) [Krauss] |
| 21112 | Empty space contains a continual flux of brief virtual particles [Krauss] |
| 21108 | The universe is precisely 13.72 billion years old [Krauss] |
| 21111 | It seems likely that cosmic inflation is eternal, and this would make a multiverse inevitable [Krauss] |
| 1742 | Thales said the gods know our wrong thoughts as well as our evil actions [Thales, by Diog. Laertius] |