Syllabus

Cardinal Arithmetic

Contents: Functions, injective and bijective functions, Schröder-Bernstein Theorem, Cantor’s theorem, proof of

|X| = |X × {0, 1}| = |X × X|

for infinite sets X, etc., cardinal arithmetic.

References:

(a) S. M. Srivastava, A Course on Borel Sets, GTM 180, Springer.

(b) S. M. Srivastava, Transfinite Numbers–What is Infinity, Resonance, V. 2, No. 3, 1997.

Set theory

Contents: Axiom of choice, ultrafilters, well orders, ordinals, transfinite induction, definition of cardinal numbers, some discussion on the independence of AC and GCH.

References:

Thomas Jech, Set Theory, Fourth Edition (2006) Springer (Part 1 in this book is a good parallel reading)

Completeness and compactness theorems in propositional and predicate logic

Contents: Propositional logic: syntax, valuations and truth, Hilbert style deduction system, soundness and completeness, strong conceptual completeness: Stone duality

Predicate logic: languages, syntax, evaluation of terms and satisfaction of formulas, a deduction system, Gödel’s completeness theorem, ultraproducts and Los’ theorem, compactness theorem.

References:

(a) S. M. Srivastava, A Course on Mathematical Logic, Second Edition, Universitext, Springer.

(b) H.B. Enderton, A Mathematical Introduction to Logic, San Diego: Harcourt, 2001.

(c) J. Bridge: Beginning Model Theory: The Completeness Theorem and Some Consequences. Oxford Logic Guides, 1977.

(d) I. Chiswell and W. Hodges: Mathematical Logic. Oxford, 2007.

(e) R. Cori and D. Lascar: Mathematical Logic, Oxford, 2001.

(f) J. Goubalt-Larrecq and J. Mackie: Proof Theory and Automated Deduction, Kluwer, 1997.

(g) J. Kelly: The Essence of Logic, Pearson, 2011.

(h) A. Margaris, First Order Mathematical Logic, Dover, 1990.

Gödel’s First Incompleteness Theorem

Contents: Recursive Functions, Gödel Numbering (Arithmetization of theories), Representability, the final step of the proof of the first incompleteness theorem.

References: S. M. Srivastava, A Course on Mathematical Logic, First or Second Edition, Universitext, Springer.