Combining Texts

All the ideas for 'works', 'Introduction to Mathematical Logic' and 'Neutral Relations'

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29 ideas

2. Reason / B. Laws of Thought / 6. Ockham's Razor
6806 Do not multiply entities beyond necessity [William of Ockham]
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
17761 A compact axiomatisation makes it possible to understand a field as a whole [Walicki]
17763 Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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]
17757 Members of ordinals are ordinals, and also subsets of ordinals [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]
8. Modes of Existence / A. Relations / 1. Nature of Relations
14217 The 'standard' view of relations is that they hold of several objects in a given order [Fine,K]
14216 The 'positionalist' view of relations says the number of places is fixed, but not the order [Fine,K]
14218 A block on top of another contains one relation, not both 'on top of' and 'beneath' [Fine,K]
14219 Language imposes a direction on a road which is not really part of the road [Fine,K]
14220 Explain biased relations as orderings of the unbiased, or the unbiased as permutation classes of the biased? [Fine,K]
8. Modes of Existence / D. Universals / 5. Universals as Concepts
22132 Species and genera are individual concepts which naturally signify many individuals [William of Ockham]
10. Modality / A. Necessity / 2. Nature of Necessity
17742 Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki]
27. Natural Reality / D. Time / 1. Nature of Time / i. Denying time
19381 The past has ceased to exist, and the future does not yet exist, so time does not exist [William of Ockham]
28. God / A. Divine Nature / 6. Divine Morality / d. God decrees morality
8010 William of Ockham is the main spokesman for God's commands being the source of morality [William of Ockham]
29. Religion / B. Monotheistic Religion / 4. Christianity / c. Angels
16679 Even an angel must have some location [William of Ockham, by Pasnau]