23 ideas
10702 | Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning [Potter] |
10713 | Usually the only reason given for accepting the empty set is convenience [Potter] |
13044 | Infinity: There is at least one limit level [Potter] |
23623 | Predicativism says only predicated sets exist [Hossack] |
10708 | Nowadays we derive our conception of collections from the dependence between them [Potter] |
23624 | The iterative conception has to appropriate Replacement, to justify the ordinals [Hossack] |
13546 | The 'limitation of size' principles say whether properties collectivise depends on the number of objects [Potter] |
23625 | Limitation of Size justifies Replacement, but then has to appropriate Power Set [Hossack] |
10707 | Mereology elides the distinction between the cards in a pack and the suits [Potter] |
10704 | We can formalize second-order formation rules, but not inference rules [Potter] |
23628 | The connective 'and' can have an order-sensitive meaning, as 'and then' [Hossack] |
23627 | 'Before' and 'after' are not two relations, but one relation with two orders [Hossack] |
10703 | Supposing axioms (rather than accepting them) give truths, but they are conditional [Potter] |
10712 | If set theory didn't found mathematics, it is still needed to count infinite sets [Potter] |
23626 | Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack] |
17882 | It is remarkable that all natural number arithmetic derives from just the Peano Axioms [Potter] |
23621 | Numbers are properties, not sets (because numbers are magnitudes) [Hossack] |
23622 | We can only mentally construct potential infinities, but maths needs actual infinities [Hossack] |
13043 | A relation is a set consisting entirely of ordered pairs [Potter] |
13042 | If dependence is well-founded, with no infinite backward chains, this implies substances [Potter] |
13041 | Collections have fixed members, but fusions can be carved in innumerable ways [Potter] |
10709 | Priority is a modality, arising from collections and members [Potter] |
3016 | Even the gods cannot strive against necessity [Pittacus, by Diog. Laertius] |