more on this theme
|
more from this text
Single Idea 13029
[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / k. Axiom of Existence
]
Full Idea
Axiom of Set Existence: ∃x (x = x). This says our universe is non-void. Under most developments of formal logic, this is derivable from the logical axioms and thus redundant, but we do so for emphasis.
Gist of Idea
Set Existence: ∃x (x = x)
Source
Kenneth Kunen (Set Theory [1980], §1.5)
Book Ref
Kunen,Kenneth: 'Set Theory: Introduction to Independence Proofs' [North-Holland 1980], p.10
The
12 ideas
from Kenneth Kunen
18465
|
An 'equivalence' relation is one which is reflexive, symmetric and transitive
[Kunen]
|
13038
|
Power Set: ∀x ∃y ∀z(z ⊂ x → z ∈ y)
[Kunen]
|
13030
|
Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y)
[Kunen]
|
13029
|
Set Existence: ∃x (x = x)
[Kunen]
|
13031
|
Comprehension: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ)
[Kunen]
|
13033
|
Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A)
[Kunen]
|
13036
|
Choice: ∀A ∃R (R well-orders A)
[Kunen]
|
13032
|
Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z)
[Kunen]
|
13034
|
Replacement: ∀x∈A ∃!y φ(x,y) → ∃Y ∀X∈A ∃y∈Y φ(x,y)
[Kunen]
|
13037
|
Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x)
[Kunen]
|
13039
|
Foundation:∀x(∃y(y∈x) → ∃y(y∈x ∧ ¬∃z(z∈x ∧ z∈y)))
[Kunen]
|
13040
|
Constructibility: V = L (all sets are constructible)
[Kunen]
|