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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 19th century arithmetization of analysis isolated the real numbers from geometry
1 p.18 Descartes showed a one-one order-preserving match between points on a line and the real numbers
1 p.19 Cantor: there is no size between naturals and reals, or between a set and its power set
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.23 'Well-ordering' must have a least member, so it does the natural numbers but not the integers
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 We can choose from finite and evident sets, but not from infinite opaque ones
1 p.27 With the Axiom of Choice every set can be well-ordered
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 Modern model theory begins with the proof of Los's Conjecture in 1962
9 p.238 Models are ways the world might be from a first-order point of view
p.350 p.350 Fregean self-evidence is an intrinsic property of basic truths, rules and definitions