Set Theory And Logic at the University of Zimbabwe

HMTH037 Set Theory And Logic

Lecturer: Mr D Vuma

Duration :2 semester

48 lectures

Aim:

The course is generally viewed as two courses `Set Theory' and `Logic' on equal weighting either running concurrently or consecutively over the two semester academic year. The overall aim is to provide the final year undergraduate mathematics specialist a general introduction to the basic ideas of Logic and axiomatic Set Theory.

Course Outline:

LOGIC: Propositional Calculus: Axioms, Deduction theorem, Completeness and consistency. 5

First order languages and first order theories: the tautology theorem, results concerning quantifiers, introduction rule, generalization rule, substitution rule, substitution theorem, distribution theorem, closure theorem, deduction theorem, theorem on constants. 9

The characterization problem: reduction theorem, reduction theorem for consistency, the completeness theorem, Lindenbaum's theorem. 6

Basic model theory: compactness theorem, L\"owenheim-Skolem's theorem and applications. 4

SET THEORY: Axiomatic foundation: Russell's paradox, axioms of set theory (extensionality, emptyset, pairset, separation, powerset, unions, and infinity axioms). 3

Revisiting classical notions: ordered pairs, Cartesian products, disjoint unions, relations, equivalence relations, classes, and partitions, functions, indexed families, structured sets. 3

Natural numbers: Peano axioms, existence, uniqueness, and recursion theorems, establishing the set \Bbb N of natural numbers along with the properties of addition, multiplication and order, well orderedness of \Bbb N. 5

Ordinals and well ordering: definitions (well order, ordinal), examples and elementary results about ordinals, ordinal arithmetic. 4

The Axiom of Choice: Hartog's lemma, axiom of replacement, axiom of choice and its controversial nature, equivalences to the axiom of choice (e.g Zermelo's Well ordering theorem, Zorn's lemma, Tychonoff's theorem), a brief discussion of foundational aspects - formalism, intuitionism, constructivism. 5

Cardinal Arithmetic: Cardinals, cardinal functions, ordering cardinals, Cantor-Bernstein theorem, the axiom of choice on classes of cardinals, Dedekind infiniteness, cardinal addition, multiplication, exponentiation and properties, continuum hypothesis and generalized continuum hypothesis. 4

Recommended Texts:

E Mendelsohn, Introduction to Mathematical Logic.
J L Krivine, Introduction to Axiomatic Set Theory.
T Jech, Axiomatic Set Theory.
J Barwise, Handbook of Mathematical Logic.

Additional Reading:

P T Johnstone, Notes on Logic and Set Theory (Cambridge University Press, 1987).
Y N Moschovakis, Notes on Set Theory (Springer-Verlag, 1994).
H D Ebbinghaus, J Flum, W Thomas, Mathematical Logic (Springer-Verlag, 1994).
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LOGIC: Propositional Calculus: Axioms, Deduction theorem, Completeness and consistency.	5
First order languages and first order theories: the tautology theorem, results concerning quantifiers, introduction rule, generalization rule, substitution rule, substitution theorem, distribution theorem, closure theorem, deduction theorem, theorem on constants.	9
The characterization problem: reduction theorem, reduction theorem for consistency, the completeness theorem, Lindenbaum's theorem.	6
Basic model theory: compactness theorem, L\"owenheim-Skolem's theorem and applications.	4
SET THEORY: Axiomatic foundation: Russell's paradox, axioms of set theory (extensionality, emptyset, pairset, separation, powerset, unions, and infinity axioms).	3
Revisiting classical notions: ordered pairs, Cartesian products, disjoint unions, relations, equivalence relations, classes, and partitions, functions, indexed families, structured sets.	3
Natural numbers: Peano axioms, existence, uniqueness, and recursion theorems, establishing the set \Bbb N of natural numbers along with the properties of addition, multiplication and order, well orderedness of \Bbb N.	5
Ordinals and well ordering: definitions (well order, ordinal), examples and elementary results about ordinals, ordinal arithmetic.	4
The Axiom of Choice: Hartog's lemma, axiom of replacement, axiom of choice and its controversial nature, equivalences to the axiom of choice (e.g Zermelo's Well ordering theorem, Zorn's lemma, Tychonoff's theorem), a brief discussion of foundational aspects - formalism, intuitionism, constructivism.	5
Cardinal Arithmetic: Cardinals, cardinal functions, ordering cardinals, Cantor-Bernstein theorem, the axiom of choice on classes of cardinals, Dedekind infiniteness, cardinal addition, multiplication, exponentiation and properties, continuum hypothesis and generalized continuum hypothesis.	4