Ideas from 'Introduction to Mathematical Logic' by Michal Walicki [2012], by Theme Structure
[found in 'Introduction to Mathematical Logic' by Walicki,Michal [World Scientific 2012,9789814343879]].
Click on the Idea Number for the full details 
back to texts

expand these ideas
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 wellordering

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. LöwenheimSkolem Theorems
17748

The LS Theorem says no theory (even of reals) says more than a natural number theory

5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17763

Axiomatic systems are purely syntactic, and do not presuppose any interpretation

17761

A compact axiomatisation makes it possible to understand a field as a whole

6. Mathematics / A. Nature of Mathematics / 3. Numbers / e. Ordinal numbers
17755

Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals

17756

The union of finite ordinals is the first 'limit ordinal'; 2ω is the second...

17757

Members of ordinals are ordinals, and also subsets 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

6. Mathematics / B. Foundations for Mathematics / 2. Axioms for Geometry
17762

In nonEuclidean 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
