Combining Texts

All the ideas for 'Introduction to Mathematical Logic', 'Essence and Potentiality' and 'Mathematics without Numbers'

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30 ideas

4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
17749 Post proved the consistency of propositional logic in 1921 [Walicki]
17765 Propositional language can only relate statements as the same or as different [Walicki]
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
17764 Boolean connectives are interpreted as functions on the set {1,0} [Walicki]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
17752 The empty set is useful for defining sets by properties, when the members are not yet known [Walicki]
17753 The empty set avoids having to take special precautions in case members vanish [Walicki]
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
17759 Ordinals play the central role in set theory, providing the model of well-ordering [Walicki]
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
17741 To determine the patterns in logic, one must identify its 'building blocks' [Walicki]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
17747 A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
17748 The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17763 Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki]
17761 A compact axiomatisation makes it possible to understand a field as a whole [Walicki]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
17757 Members of ordinals are ordinals, and also subsets of ordinals [Walicki]
17758 Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki]
17755 Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki]
17756 The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki]
17760 Two infinite ordinals can represent a single infinite cardinal [Walicki]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
17762 In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
17754 Inductive proof depends on the choice of the ordering [Walicki]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
8698 Modal structuralism says mathematics studies possible structures, which may or may not be actualised [Hellman, by Friend]
9557 Statements of pure mathematics are elliptical for a sort of modal conditional [Hellman, by Chihara]
10263 Modal structuralism can only judge possibility by 'possible' models [Shapiro on Hellman]
8. Modes of Existence / C. Powers and Dispositions / 4. Powers as Essence
17954 Essence is a thing's necessities, but what about its possibilities (which may not be realised)? [Vetter]
9. Objects / D. Essence of Objects / 4. Essence as Definition
17953 Real definition fits abstracta, but not individual concrete objects like Socrates [Vetter]
9. Objects / D. Essence of Objects / 7. Essence and Necessity / a. Essence as necessary properties
17952 Modal accounts make essence less mysterious, by basing them on the clearer necessity [Vetter]
10. Modality / A. Necessity / 2. Nature of Necessity
17742 Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki]
10. Modality / A. Necessity / 5. Metaphysical Necessity
17959 Metaphysical necessity is even more deeply empirical than Kripke has argued [Vetter]
10. Modality / B. Possibility / 1. Possibility
17955 Possible worlds allow us to talk about degrees of possibility [Vetter]
17957 Maybe possibility is constituted by potentiality [Vetter]
10. Modality / D. Knowledge of Modality / 4. Conceivable as Possible / c. Possible but inconceivable
17958 The apparently metaphysically possible may only be epistemically possible [Vetter]
10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
17956 Closeness of worlds should be determined by the intrinsic nature of relevant objects [Vetter]