Combining Texts
Ideas for
'The Evolution of Logic', 'Explaining the A Priori' and 'Dispositions and Powers'
expand these ideas
|
start again
|
choose
another area for these texts
display all the ideas for this combination of texts
16 ideas
4. Formal Logic / F. Set Theory ST / 1. Set Theory
|
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 / i. Axiom of Foundation VIII
|
13493
|
In the modern view, foundation is the heart of the way to do set theory [Hart,WD]
|
|
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 / 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]
|