Combining Texts

All the ideas for 'Set Theory', 'Two-Dimensional Semantics' and 'Mathematical Explanation'

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33 ideas

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
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]
9. Objects / D. Essence of Objects / 3. Individual Essences
13230 Particular essence is often captured by generality [Steiner,M]
10. Modality / A. Necessity / 3. Types of Necessity
14703 Superficial necessity is true in all worlds; deep necessity is thus true, no matter which world is actual [Schroeter]
10. Modality / D. Knowledge of Modality / 4. Conceivable as Possible / b. Conceivable but impossible
14714 Contradictory claims about a necessary god both seem apriori coherent [Schroeter]
12. Knowledge Sources / A. A Priori Knowledge / 8. A Priori as Analytic
14704 2D semantics gives us apriori knowledge of our own meanings [Schroeter]
14. Science / D. Explanation / 2. Types of Explanation / e. Lawlike explanations
13229 Maybe an instance of a generalisation is more explanatory than the particular case [Steiner,M]
14. Science / D. Explanation / 2. Types of Explanation / m. Explanation by proof
13231 Explanatory proofs rest on 'characterizing properties' of entities or structure [Steiner,M]
18. Thought / C. Content / 5. Twin Earth
14706 Your view of water depends on whether you start from the actual Earth or its counterfactual Twin [Schroeter]
18. Thought / C. Content / 7. Narrow Content
14711 Rationalists say knowing an expression is identifying its extension using an internal cognitive state [Schroeter]
19. Language / A. Nature of Meaning / 1. Meaning
14717 Internalist meaning is about understanding; externalist meaning is about embedding in a situation [Schroeter]
19. Language / C. Assigning Meanings / 2. Semantics
14720 Semantic theory assigns meanings to expressions, and metasemantics explains how this works [Schroeter]
19. Language / C. Assigning Meanings / 4. Compositionality
14695 Semantic theories show how truth of sentences depends on rules for interpreting and joining their parts [Schroeter]
19. Language / C. Assigning Meanings / 7. Extensional Semantics
14697 'Federer' and 'best tennis player' can't mean the same, despite having the same extension [Schroeter]
14696 Simple semantics assigns extensions to names and to predicates [Schroeter]
19. Language / C. Assigning Meanings / 8. Possible Worlds Semantics
14698 Possible worlds semantics uses 'intensions' - functions which assign extensions at each world [Schroeter]
14699 Possible worlds make 'I' and that person's name synonymous, but they have different meanings [Schroeter]
14709 Possible worlds semantics implies a constitutive connection between meanings and modal claims [Schroeter]
14719 In the possible worlds account all necessary truths are same (because they all map to the True) [Schroeter]
19. Language / C. Assigning Meanings / 10. Two-Dimensional Semantics
14701 Array worlds along the horizontal, and contexts (world,person,time) along the vertical [Schroeter]
14702 If we introduce 'actually' into modal talk, we need possible worlds twice to express this [Schroeter]
14705 Do we know apriori how we refer to names and natural kinds, but their modal profiles only a posteriori? [Schroeter]
14715 2D fans defend it for conceptual analysis, for meaning, and for internalist reference [Schroeter]
14716 2D semantics can't respond to contingent apriori claims, since there is no single proposition involved [Schroeter]