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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 13457 A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets
1 p.23 13458 A partial ordering becomes 'total' if any two members of its field are comparable
1 p.23 13460 'Well-ordering' must have a least member, so it does the natural numbers but not the integers
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 13461 We can choose from finite and evident sets, but not from infinite opaque ones
1 p.27 13462 With the Axiom of Choice every set can be well-ordered
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 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