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
                        Full Idea: A proof of the consistency of propositional logic was given by Emil Post in 1921.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2.1)
Propositional language can only relate statements as the same or as different
                        Full Idea: Propositional language is very rudimentary and has limited powers of expression. The only relation between various statements it can handle is that of identity and difference. As are all the same, but Bs can be different from As.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], 7 Intro)
                        A reaction: [second sentence a paraphrase] In predicate logic you could represent two statements as being the same except for one element (an object or predicate or relation or quantifier).
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
Boolean connectives are interpreted as functions on the set {1,0}
                        Full Idea: Boolean connectives are interpreted as functions on the set {1,0}.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], 5.1)
                        A reaction: 1 and 0 are normally taken to be true (T) and false (F). Thus the functions output various combinations of true and false, which are truth tables.
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
                        Full Idea: The empty set is mainly a mathematical convenience - defining a set by describing the properties of its members in an involved way, we may not know from the very beginning what its members are.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1)
The empty set avoids having to take special precautions in case members vanish
                        Full Idea: Without the assumption of the empty set, one would often have to take special precautions for the case where a set happened to contain no elements.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1)
                        A reaction: Compare the introduction of the concept 'zero', where special precautions are therefore required. ...But other special precautions are needed without zero. Either he pays us, or we pay him, or ...er. Intersecting sets need the empty set.
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
                        Full Idea: Ordinals play the central role in set theory, providing the paradigmatic well-orderings.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3)
                        A reaction: When you draw the big V of the iterative hierarchy of sets (built from successive power sets), the ordinals are marked as a single line up the middle, one ordinal for each level.
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
To determine the patterns in logic, one must identify its 'building blocks'
                        Full Idea: In order to construct precise and valid patterns of arguments one has to determine their 'building blocks'. One has to identify the basic terms, their kinds and means of combination.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], History Intro)
                        A reaction: A deceptively simple and important idea. All explanation requires patterns and levels, and it is the idea of building blocks which makes such things possible. It is right at the centre of our grasp of everything.
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
                        Full Idea: A specification of a domain of objects, and of the rules for interpreting the symbols of a logical language in this domain such that all the theorems of the logical theory are true is said to be a 'model' of the theory.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.1.3)
                        A reaction: The basic ideas of this emerged 1915-30, but it needed Tarski's account of truth to really get it going.
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
                        Full Idea: The L-S Theorem is ...a shocking result, since it implies that any consistent formal theory of everything - even about biology, physics, sets or the real numbers - can just as well be understood as being about natural numbers. It says nothing more.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2)
                        A reaction: Illuminating. Particularly the point that no theory about the real numbers can say anything more than a theory about the natural numbers. So the natural numbers contain all the truths we can ever express? Eh?????
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
A compact axiomatisation makes it possible to understand a field as a whole
                        Full Idea: Having such a compact [axiomatic] presentation of a complicated field [such as Euclid's], makes it possible to relate not only to particular theorems but also to the whole field as such.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1)
Axiomatic systems are purely syntactic, and do not presuppose any interpretation
                        Full Idea: Axiomatic systems, their primitive terms and proofs, are purely syntactic, that is, do not presuppose any interpretation. ...[142] They never address the world directly, but address a possible semantic model which formally represents the world.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1)
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
Members of ordinals are ordinals, and also subsets of ordinals
                        Full Idea: Every member of an ordinal is itself an ordinal, and every ordinal is a transitive set (its members are also its subsets; a member of a member of an ordinal is also a member of the ordinal).
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3)
Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion
                        Full Idea: An ordinal can be defined as a transitive set of transitive sets, or else, as a transitive set totally ordered by set inclusion.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3)
Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals
                        Full Idea: The collection of ordinals is defined inductively: Basis: the empty set is an ordinal; Ind: for an ordinal x, the union with its singleton is also an ordinal; and any arbitrary (possibly infinite) union of ordinals is an ordinal.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3)
                        A reaction: [symbolism translated into English] Walicki says they are called 'ordinal numbers', but are in fact a set.
The union of finite ordinals is the first 'limit ordinal'; 2ω is the second...
                        Full Idea: We can form infinite ordinals by taking unions of ordinals. We can thus form 'limit ordinals', which have no immediate predecessor. ω is the first (the union of all finite ordinals), ω + ω = sω is second, 3ω the third....
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3)
Two infinite ordinals can represent a single infinite cardinal
                        Full Idea: There may be several ordinals for the same cardinality. ...Two ordinals can represent different ways of well-ordering the same number (aleph-0) of elements.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3)
                        A reaction: This only applies to infinite ordinals and cardinals. For the finite, the two coincide. In infinite arithmetic the rules are different.
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate
                        Full Idea: Since non-Euclidean geometry preserves all Euclid's postulates except the fifth one, all the theorems derived without the use of the fifth postulate remain valid.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1)
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
Inductive proof depends on the choice of the ordering
                        Full Idea: Inductive proof is not guaranteed to work in all cases and, particularly, it depends heavily on the choice of the ordering.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.1.1)
                        A reaction: There has to be an well-founded ordering for inductive proofs to be possible.
10. Modality / A. Necessity / 2. Nature of Necessity
Scotus based modality on semantic consistency, instead of on what the future could allow
                        Full Idea: The link between time and modality was severed by Duns Scotus, who proposed a notion of possibility based purely on the notion of semantic consistency. 'Possible' means for him logically possible, that is, not involving contradiction.
                        From: Michal Walicki (Introduction to Mathematical Logic [2012], History B.4)