Combining Texts

Ideas for 'The Evolution of Logic', '03: Book of Leviticus' and 'Investigations in the Foundations of Set Theory I'

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25 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory
17608 We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo]
17607 Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo]
13456 Set theory articulates the concept of order (through relations) [Hart,WD]
13497 Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe [Hart,WD]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / a. Symbols of ST
13443 ∈ relates across layers, while ⊆ relates within layers [Hart,WD]
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 [Hart,WD]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
10870 ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Zermelo, by Clegg]
13012 Zermelo published his axioms in 1908, to secure a controversial proof [Zermelo, by Maddy]
17609 Set theory can be reduced to a few definitions and seven independent axioms [Zermelo]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
13017 Zermelo introduced Pairing in 1930, and it seems fairly obvious [Zermelo, by Maddy]
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 [Hart,WD]
13015 Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy]
13495 Foundation Axiom: an nonempty set has a member disjoint from it [Hart,WD]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
13461 We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD]
13462 With the Axiom of Choice every set can be well-ordered [Hart,WD]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / m. Axiom of Separation
13486 Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Zermelo, by Hart,WD]
13020 The Axiom of Separation requires set generation up to one step back from contradiction [Zermelo, by Maddy]
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 [Hart,WD]
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 [Hart,WD]
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 [Hart,WD]
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 [Hart,WD]
13458 A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD]
13460 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD]
13490 Von Neumann defines α<β as α∈β [Hart,WD]
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 [Hart,WD]