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Single Idea 13039
[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
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Full Idea
Axiom of Foundation: ∀x (∃y(y ∈ x) → ∃y(y ∈ x ∧ ¬∃z(z ∈ x ∧ z ∈ y))). Aka the 'Axiom of Regularity'. Combined with Choice, it means there are no downward infinite chains.
Gist of Idea
Foundation:∀x(∃y(y∈x) → ∃y(y∈x ∧ ¬∃z(z∈x ∧ z∈y)))
Source
Kenneth Kunen (Set Theory [1980], §3.4)
Book Ref
Kunen,Kenneth: 'Set Theory: Introduction to Independence Proofs' [North-Holland 1980], p.100
The
12 ideas
from Kenneth Kunen
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18465
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An 'equivalence' relation is one which is reflexive, symmetric and transitive
[Kunen]
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13038
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Power Set: ∀x ∃y ∀z(z ⊂ x → z ∈ y)
[Kunen]
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13030
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Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y)
[Kunen]
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13029
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Set Existence: ∃x (x = x)
[Kunen]
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13031
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Comprehension: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ)
[Kunen]
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13034
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Replacement: ∀x∈A ∃!y φ(x,y) → ∃Y ∀X∈A ∃y∈Y φ(x,y)
[Kunen]
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13032
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Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z)
[Kunen]
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13033
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Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A)
[Kunen]
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13036
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Choice: ∀A ∃R (R well-orders A)
[Kunen]
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13037
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Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x)
[Kunen]
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13039
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Foundation:∀x(∃y(y∈x) → ∃y(y∈x ∧ ¬∃z(z∈x ∧ z∈y)))
[Kunen]
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13040
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Constructibility: V = L (all sets are constructible)
[Kunen]
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