Combining Texts
Ideas for
'Philosophical Explanations', 'Set Theory' and 'Introduction to the Theory of Logic'
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display all the ideas for this combination of texts
16 ideas
4. Formal Logic / F. Set Theory ST / 1. Set Theory
10888
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Sets can be defined by 'enumeration', or by 'abstraction' (based on a property) [Zalabardo]
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4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
10889
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The 'Cartesian Product' of two sets relates them by pairing every element with every element [Zalabardo]
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10890
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A 'partial ordering' is reflexive, antisymmetric and transitive [Zalabardo]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
10886
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Determinacy: an object is either in a set, or it isn't [Zalabardo]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
13030
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Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y) [Kunen]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
13032
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Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z) [Kunen]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
13033
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Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A) [Kunen]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
13037
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Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x) [Kunen]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
13038
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Power Set: ∀x ∃y ∀z(z ⊂ x → z ∈ y) [Kunen]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
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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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))) [Kunen]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
13036
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Choice: ∀A ∃R (R well-orders A) [Kunen]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / k. Axiom of Existence
13029
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Set Existence: ∃x (x = x) [Kunen]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / l. Axiom of Specification
10887
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Specification: Determinate totals of objects always make a set [Zalabardo]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension
13031
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Comprehension: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ) [Kunen]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / o. Axiom of Constructibility V = L
13040
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Constructibility: V = L (all sets are constructible) [Kunen]
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