HS 442 : Logic and Foundations of Mathematics

History of the relation between logic and mathematics.  Geometry and the axiomatic nature of mathematics.   Role of logic and mathematics in science.

Syntax and semantics: formal systems. Example of first order language.  Constructive problems in the notions of truth, model, consistency and completeness.  Constructive criticisms of Godel's proof primitive Recursive Functions.

Foundations of Number theory, Axiomatic and constructive approaches, Cantorian set theory and paradoxes.  The problem of infinity, Mathematical Induction, Infinite sets.

Brief survey of Platonism, Logicism, Formalism, Intuitionism, Conventionalism.  Limitations of the formalist foundations and computability, Turing Machines, Markov Algorithms and Recursion theory.

Meaning and existence in mathematics; views of mathematicians and philosophers.  Examples of constructive results.

Texts/References

G.K. Kneebone, Mathematical Logic and the Foundations of Mathematics: An Introductory Survey, Van Nostrand, 1963.

Delong, Howard, A Profile of Mathematical Logic, Addison Wesle Publishing Co., 1971.

R.L. Wilder, Introduction to the Foundations of Mathematics, Second edition, John Wiley, 1965.

S.C. Kleene, Introduction to Meta-Mathematics, Van Nostrand, 1952.