Ideas of Michal Walicki, by Theme

[Norwegian, fl. 2012, At the University of Bergen, Norway.]

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4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
 17749 Post proved the consistency of propositional logic in 1921
 17765 Propositional language can only relate statements as the same or as different
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
 17764 Boolean connectives are interpreted as functions on the set {1,0}
4. Formal Logic / C. Predicate Calculus PC / 3. Completeness of PC
 17751 Gödel proved the completeness of first order predicate logic in 1930
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
 17753 The empty set avoids having to take special precautions in case members vanish
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
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'
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
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
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
 17761 A compact axiomatisation makes it possible to understand a field as a whole
 17763 Axiomatic systems are purely syntactic, and do not presuppose any interpretation
6. Mathematics / A. Nature of Mathematics / 3. Numbers / e. Ordinal numbers
 17756 The union of finite ordinals is the first 'limit ordinal'; 2ω is the second...
 17755 Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals
 17758 Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion
 17760 Two infinite ordinals can represent a single infinite cardinal
 17757 Members of ordinals are ordinals, and also subsets of ordinals
6. Mathematics / B. Foundations for Mathematics / 2. Axioms for Geometry
 17762 In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / f. Mathematical induction
 17754 Inductive proof depends on the choice of the ordering
10. Modality / A. Necessity / 2. Nature of Necessity
 17742 Scotus based modality on semantic consistency, instead of on what the future could allow