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Ideas of William D. Hart, by Text

[American, fl. 1994, At the University of Illinois, Chicago.]

1992 Hat-Tricks and Heaps
p.2 9117 The smallest heap has four objects: three on the bottom, one on the top [Sorensen]
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.18 13446 19th century arithmetization of analysis isolated the real numbers from geometry
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
2 p.31 13466 We are all post-Kantians, because he set the current agenda for philosophy
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.71 13488 Mass words do not have plurals, or numerical adjectives, or use 'fewer'
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 13496 First-order logic is 'compact': consequences of a set are consequences of a finite subset
3 p.80 13494 The iterative conception may not be necessary, and may have fixed points or infinitely descending chains
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 13512 Modern model theory begins with the proof of Los's Conjecture in 1962
9 p.238 13513 Models are ways the world might be from a first-order point of view
p.350 p.350 13480 Fregean self-evidence is an intrinsic property of basic truths, rules and definitions