Ideas of William D. Hart, by Theme

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

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1. Philosophy / C. History of Philosophy / 4. Later European Philosophy / c. Eighteenth century philosophy
13466 We are all post-Kantians, because he set the current agenda for philosophy
1. Philosophy / D. Nature of Philosophy / 5. Aims of Philosophy / d. Philosophy as puzzles
13477 The problems are the monuments of philosophy
1. Philosophy / F. Analytic Philosophy / 6. Logical Analysis
13515 To study abstract problems, some knowledge of set theory is essential
3. Truth / C. Correspondence Truth / 2. Correspondence to Facts
13469 Tarski showed how we could have a correspondence theory of truth, without using 'facts'
3. Truth / F. Semantic Truth / 1. Tarski's Truth / b. Satisfaction and truth
13504 Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do
3. Truth / F. Semantic Truth / 2. Semantic Truth
13503 A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
13500 Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃
13502 ∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...'
4. Formal Logic / F. Set Theory ST / 1. Set Theory
13456 Set theory articulates the concept of order (through relations)
13497 Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / a. Symbols of ST
13443 ∈ relates across layers, while ⊆ relates within layers
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
13442 Without the empty set we could not form a∩b without checking that a and b meet
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
13493 In the modern view, foundation is the heart of the way to do set theory
13495 Foundation Axiom: an nonempty set has a member disjoint from it
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
13462 With the Axiom of Choice every set can be well-ordered
13461 We can choose from finite and evident sets, but not from infinite opaque ones
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / o. Axiom of Constructibility V = L
13516 If we accept that V=L, it seems to settle all the open questions of set theory
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
13441 Naïve set theory has trouble with comprehension, the claim that every predicate has an extension
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
13494 The iterative conception may not be necessary, and may have fixed points or infinitely descending chains
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
13457 A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets
13460 'Well-ordering' must have a least member, so it does the natural numbers but not the integers
13458 A partial ordering becomes 'total' if any two members of its field are comparable
13490 Von Neumann defines α<β as α∈β
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
13481 Maybe sets should be rethought in terms of the even more basic categories
5. Theory of Logic / G. Quantification / 3. Objectual Quantification
13506 The universal quantifier can't really mean 'all', because there is no universal set
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
13505 Model theory studies how set theory can model sets of sentences
13511 Model theory is mostly confined to first-order theories
13513 Models are ways the world might be from a first-order point of view
13512 Modern model theory begins with the proof of Los's Conjecture in 1962
5. Theory of Logic / K. Features of Logics / 6. Compactness
13496 First-order logic is 'compact': consequences of a set are consequences of a finite subset
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
13484 Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / c. Burali-Forti's paradox
13482 The Burali-Forti paradox is a crisis for Cantor's ordinals
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox
13507 The machinery used to solve the Liar can be rejigged to produce a new Liar
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / b. The Heap paradox ('Sorites')
9117 The smallest heap has four objects: three on the bottom, one on the top [Sorensen]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13459 The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers
13463 There are at least as many infinite cardinals as transfinite ordinals (because they will map)
13491 The axiom of infinity with separation gives a least limit ordinal ω
13492 Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
13446 19th century arithmetization of analysis isolated the real numbers from geometry
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
13509 We can establish truths about infinite numbers by means of induction
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
13474 Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
13471 Mathematics makes existence claims, but philosophers usually say those are never analytic
7. Existence / C. Structure of Existence / 8. Stuff / a. Pure stuff
13488 Mass words do not have plurals, or numerical adjectives, or use 'fewer'
12. Knowledge Sources / A. A Priori Knowledge / 2. Self-Evidence
13480 Fregean self-evidence is an intrinsic property of basic truths, rules and definitions
12. Knowledge Sources / A. A Priori Knowledge / 11. Denying the A Priori
13476 The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori
18. Thought / D. Concepts / 3. Ontology of Concepts / c. Fregean concepts
13475 The Fregean concept of GREEN is a function assigning true to green things, and false to the rest