11 ideas
17833 | The first-order ZF axiomatisation is highly non-categorical [Hallett,M] |
17834 | Non-categoricity reveals a sort of incompleteness, with sets existing that the axioms don't reveal [Hallett,M] |
15717 | Using Choice, you can cut up a small ball and make an enormous one from the pieces [Kaplan/Kaplan] |
17837 | Zermelo allows ur-elements, to enable the widespread application of set-theory [Hallett,M] |
15712 | 1 and 0, then add for naturals, subtract for negatives, divide for rationals, take roots for irrationals [Kaplan/Kaplan] |
15711 | The rationals are everywhere - the irrationals are everywhere else [Kaplan/Kaplan] |
15714 | 'Commutative' laws say order makes no difference; 'associative' laws say groupings make no difference [Kaplan/Kaplan] |
15715 | 'Distributive' laws say if you add then multiply, or multiply then add, you get the same result [Kaplan/Kaplan] |
17836 | The General Continuum Hypothesis and its negation are both consistent with ZF [Hallett,M] |
3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
15713 | The first million numbers confirm that no number is greater than a million [Kaplan/Kaplan] |