Ideas from 'Set Theory' by Kenneth Kunen [1980], by Theme Structure
[found in 'Set Theory: Introduction to Independence Proofs' by Kunen,Kenneth [North-Holland 1980,0-444-85-401-0]].
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
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Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y)
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Full Idea:
Axiom of Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y). That is, a set is determined by its members. If every z in one set is also in the other set, then the two sets are the same.
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From:
Kenneth Kunen (Set Theory [1980], §1.5)
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
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Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z)
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Full Idea:
Axiom of Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z). Any pair of entities must form a set.
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From:
Kenneth Kunen (Set Theory [1980], §1.6)
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A reaction:
Repeated applications of this can build the hierarchy of sets.
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
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Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A)
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Full Idea:
Axiom of Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A). That is, the union of a set (all the members of the members of the set) must also be a set.
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From:
Kenneth Kunen (Set Theory [1980], §1.6)
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
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Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x)
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Full Idea:
Axiom of Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x). That is, there is a set which contains zero and all of its successors, hence all the natural numbers. The principal of induction rests on this axiom.
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From:
Kenneth Kunen (Set Theory [1980], §1.7)
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
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Power Set: ∀x ∃y ∀z(z ⊂ x → z ∈ y)
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Full Idea:
Power Set Axiom: ∀x ∃y ∀z(z ⊂ x → z ∈ y). That is, there is a set y which contains all of the subsets of a given set. Hence we define P(x) = {z : z ⊂ x}.
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From:
Kenneth Kunen (Set Theory [1980], §1.10)
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
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Replacement: ∀x∈A ∃!y φ(x,y) → ∃Y ∀X∈A ∃y∈Y φ(x,y)
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Full Idea:
Axiom of Replacement Scheme: ∀x ∈ A ∃!y φ(x,y) → ∃Y ∀X ∈ A ∃y ∈ Y φ(x,y). That is, any function from a set A will produce another set Y.
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From:
Kenneth Kunen (Set Theory [1980], §1.6)
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
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Foundation:∀x(∃y(y∈x) → ∃y(y∈x ∧ ¬∃z(z∈x ∧ z∈y)))
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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.
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From:
Kenneth Kunen (Set Theory [1980], §3.4)
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
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Choice: ∀A ∃R (R well-orders A)
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Full Idea:
Axiom of Choice: ∀A ∃R (R well-orders A). That is, for every set, there must exist another set which imposes a well-ordering on it. There are many equivalent versions. It is not needed in elementary parts of set theory.
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From:
Kenneth Kunen (Set Theory [1980], §1.6)
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / k. Axiom of Existence
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Set Existence: ∃x (x = x)
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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.
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From:
Kenneth Kunen (Set Theory [1980], §1.5)
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension
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Comprehension: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ)
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Full Idea:
Comprehension Scheme: for each formula φ without y free, the universal closure of this is an axiom: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ). That is, there must be a set y if it can be defined by the formula φ.
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From:
Kenneth Kunen (Set Theory [1980], §1.5)
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A reaction:
Unrestricted comprehension leads to Russell's paradox, so restricting it in some way (e.g. by the Axiom of Specification) is essential.
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / o. Axiom of Constructibility V = L
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Constructibility: V = L (all sets are constructible)
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Full Idea:
Axiom of Constructability: this is the statement V = L (i.e. ∀x ∃α(x ∈ L(α)). That is, the universe of well-founded von Neumann sets is the same as the universe of sets which are actually constructible. A possible axiom.
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From:
Kenneth Kunen (Set Theory [1980], §6.3)
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