1992 | Hat-Tricks and Heaps |
p.2 | 9117 | The smallest heap has four objects: three on the bottom, one on the top |
2010 | The Evolution of Logic |
p.22 | 13456 | Set theory articulates the concept of order (through relations) |
1 | p.4 | 13442 | Without the empty set we could not form a∩b without checking that a and b meet |
1 | p.4 | 13441 | Naïve set theory has trouble with comprehension, the claim that every predicate has an extension |
1 | p.5 | 13443 | ∈ relates across layers, while ⊆ relates within layers |
1 | p.16 | 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x |
1 | p.18 | 13446 | 19th century arithmetization of analysis isolated the real numbers from geometry |
1 | p.18 | 13445 | Descartes showed a one-one order-preserving match between points on a line and the real numbers |
1 | p.19 | 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set |
1 | p.23 | 13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers |
1 | p.23 | 13458 | A partial ordering becomes 'total' if any two members of its field are comparable |
1 | p.23 | 13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets |
1 | p.26 | 13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers |
1 | p.27 | 13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) |
1 | p.27 | 13462 | With the Axiom of Choice every set can be well-ordered |
1 | p.27 | 13461 | We can choose from finite and evident sets, but not from infinite opaque ones |
10 | p.268 | 13515 | To study abstract problems, some knowledge of set theory is essential |
10 | p.270 | 13516 | If we accept that V=L, it seems to settle all the open questions of set theory |
10 | p.273 | 13517 | If set theory is consistent, we cannot refute or prove the Continuum Hypothesis |
2 | p.31 | 13466 | We are all post-Kantians, because he set the current agenda for philosophy |
2 | p.32 | 13467 | Leibniz was the first modern to focus on sentence-sized units (where empiricists preferred word-size) |
2 | p.36 | 13469 | Tarski showed how we could have a correspondence theory of truth, without using 'facts' |
2 | p.41 | 13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic |
2 | p.44 | 13474 | Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several |
2 | p.47 | 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest |
2 | p.53 | 13476 | The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori |
2 | p.53 | 13477 | The problems are the monuments of philosophy |
2 | p.58 | 13481 | Maybe sets should be rethought in terms of the even more basic categories |
3 | p.59 | 13482 | The Burali-Forti paradox is a crisis for Cantor's ordinals |
3 | p.63 | 13484 | Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that |
3 | p.69 | 13486 | Not every predicate has an extension, but Separation picks the members that satisfy a predicate |
3 | p.71 | 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' |
3 | p.73 | 13489 | Von Neumann treated cardinals as a special sort of ordinal |
3 | p.74 | 13491 | The axiom of infinity with separation gives a least limit ordinal ω |
3 | p.74 | 13490 | Von Neumann defines α<β as α∈β |
3 | p.75 | 13492 | Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton |
3 | p.79 | 13493 | In the modern view, foundation is the heart of the way to do set theory |
3 | p.80 | 13495 | Foundation Axiom: an nonempty set has a member disjoint from it |
3 | p.80 | 13494 | The iterative conception may not be necessary, and may have fixed points or infinitely descending chains |
3 | p.80 | 13496 | First-order logic is 'compact': consequences of a set are consequences of a finite subset |
3 | p.88 | 13497 | Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe |
4 | p.90 | 13500 | Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent |
4 | p.96 | 13502 | ∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...' |
4 | p.101 | 13503 | A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth |
4 | p.107 | 13504 | Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do |
4 | p.108 | 13505 | Model theory studies how set theory can model sets of sentences |
4 | p.111 | 13506 | The universal quantifier can't really mean 'all', because there is no universal set |
4 | p.122 | 13507 | The machinery used to solve the Liar can be rejigged to produce a new Liar |
5 | p.144 | 13509 | We can establish truths about infinite numbers by means of induction |
9 | p.236 | 13511 | Model theory is mostly confined to first-order theories |
9 | p.238 | 13513 | Models are ways the world might be from a first-order point of view |
9 | p.238 | 13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 |
p.350 | p.350 | 13480 | Fregean self-evidence is an intrinsic property of basic truths, rules and definitions |