Combining Philosophers
Ideas for Anaxarchus, George Boolos and Kenneth Kunen
expand these ideas
|
start again
|
choose
another area for these philosophers
display all the ideas for this combination of philosophers
19 ideas
4. Formal Logic / F. Set Theory ST / 1. Set Theory
|
10482
|
The logic of ZF is classical first-order predicate logic with identity [Boolos]
|
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
|
10492
|
A few axioms of set theory 'force themselves on us', but most of them don't [Boolos]
|
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
|
13030
|
Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y) [Kunen]
|
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
|
13032
|
Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z) [Kunen]
|
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
|
13033
|
Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A) [Kunen]
|
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
|
13037
|
Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x) [Kunen]
|
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
|
13038
|
Power Set: ∀x ∃y ∀z(z ⊂ x → z ∈ y) [Kunen]
|
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
|
18192
|
Do the Replacement Axioms exceed the iterative conception of sets? [Boolos, by Maddy]
|
|
13034
|
Replacement: ∀x∈A ∃!y φ(x,y) → ∃Y ∀X∈A ∃y∈Y φ(x,y) [Kunen]
|
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
|
13039
|
Foundation:∀x(∃y(y∈x) → ∃y(y∈x ∧ ¬∃z(z∈x ∧ z∈y))) [Kunen]
|
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
|
13036
|
Choice: ∀A ∃R (R well-orders A) [Kunen]
|
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / k. Axiom of Existence
|
13029
|
Set Existence: ∃x (x = x) [Kunen]
|
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension
|
13031
|
Comprehension: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ) [Kunen]
|
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / o. Axiom of Constructibility V = L
|
13040
|
Constructibility: V = L (all sets are constructible) [Kunen]
|
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
|
7785
|
The use of plurals doesn't commit us to sets; there do not exist individuals and collections [Boolos]
|
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
|
10485
|
Naïve sets are inconsistent: there is no set for things that do not belong to themselves [Boolos]
|
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
|
10484
|
The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first [Boolos]
|
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
|
13547
|
Limitation of Size is weak (Fs only collect is something the same size does) or strong (fewer Fs than objects) [Boolos, by Potter]
|
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
|
10699
|
Does a bowl of Cheerios contain all its sets and subsets? [Boolos]
|