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