Ideas from 'Introduction to Mathematical Logic' by Michal Walicki [2012], by Theme Structure

[found in 'Introduction to Mathematical Logic' by Walicki,Michal [World Scientific 2012,978-981-4343-87-9]].

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4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
Post proved the consistency of propositional logic in 1921
Propositional language can only relate statements as the same or as different
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
Boolean connectives are interpreted as functions on the set {1,0}
4. Formal Logic / C. Predicate Calculus PC / 3. Completeness of PC
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
The empty set is useful for defining sets by properties, when the members are not yet known
The empty set avoids having to take special precautions in case members vanish
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
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
To determine the patterns in logic, one must identify its 'building blocks'
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
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
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
A compact axiomatisation makes it possible to understand a field as a whole
Axiomatic systems are purely syntactic, and do not presuppose any interpretation
6. Mathematics / A. Nature of Mathematics / 3. Numbers / e. Ordinal numbers
The union of finite ordinals is the first 'limit ordinal'; 2ω is the second...
Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion
Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals
Two infinite ordinals can represent a single infinite cardinal
Members of ordinals are ordinals, and also subsets of ordinals
6. Mathematics / B. Foundations for Mathematics / 2. Axioms for Geometry
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
Inductive proof depends on the choice of the ordering
10. Modality / A. Necessity / 2. Nature of Necessity
Scotus based modality on semantic consistency, instead of on what the future could allow