STATE UNIVERSITY COLLEGE AT BUFFALO

Department of Mathematics

Request for Course

I. Number and title of course

Mathematics 631, Foundations of Mathematics
II. Reason for addition to the present curricula
To present to students in mathematics and philosophy an opportunity to study topics related to the fundamental nature and foundations of mathematics.
III. Some major objectives of the course
A. To develop and study some of the properties that are characteristic of the natural numbers and the real numbers; for example, the Feano axioms, the properties that the rational numbers are both countably infinite and dense in the real numbers.

B. To study some set theory; especially as related to the natural numbers and the real numbers. To study the topic of transfinite induction.

C. To examine and explain some of the different branches of mathematics as "otology, number theory, geometry, algebra and logic. To mention some of the significant problems that arise in these different areas, and hour these problems may have influenced the trends in mathematics.

D. To introduce to the student some of the important people that played a role in the development of mathematics, by mentioning some of the problems and works associated with them; for example, Euclid, Cantor, Riemann, Fermat, Newton, Godel, and Lebesgue.

E. To explore the development of transfinite numbers.

F. To study various mathematical topics such as axiomatic systems, formal theories, consistency and independence problems, decision procedures, constructive proofs, etc.

IV. Topical outline
A. Natural and real numbers
1. Peano axioms
2. Finite and infinite sets
3. Prime numbers and some fundamental properties about primes, unsolved problems Rational numbers, irrational numbers, and the density of rational and irrational numbers
B. Elementary set theory
1. Countably infinite sets and uncountable sets
2. Transfinite induction
3. Cantor-Bernstein-Schroder theorem
C. Underlying aims and interests of some particular branches of mathematics
1. Algebra
2. Number theory
3. Topology
4. Geometry
5. Logic
6. Analysis
D. Metamathematics
1. Informal theories in context of set theory
2. Formal theories
3. Consistency and independence
4. Decision Problems
V. Texts
Hatcher, W.S. Foundation of Mathematics. Saunders Co., Philadelphia, 1968.
Kamke, E. The Theory of Sets. Dover Publishers, 1950.
Stoll, R.R. Set Theory and Logic. Freeman Co., San Francisco, 1963.
Wilder, R. L. The Foundation of Mathematics. John Wiley & Sons, Inc. 1952
VI. Prerequisites
At least two semester courses in mathematics beyond the Calculus sequence.
VII. Credit
Three semester hours
VIII. Statement of Approval
This course proposal was examined in accordance with recommended procedures and was approved by the Graduate Faculty of mathematics.
_______________________________________Chairman
IX. Catalog description
Mathematics 631 - FOUNDATIONS OF MATHEMATICS

The Axiomatic Methods fundamental properties of the natural numbers and real numbers, finite and transfinite induction; underlying aims and techniques of some different branches of mathematics.

Classification: Elective for graduate mathematics majors

Prerequisite: At least two courses in mathematics beyond the three semester sequence in Calculus.

Credit: Three semester hours

X. Statement of qualifications of faculty who will teach course
NAME PREPARATION EXPERIENCE
Barback, Joseph Ph.D. in mathematics received from Rutgers University in 1964. Research interests include Mathematical Logic and Recursive Function theory.

Barr, Jack Course work completed including research 6 years experience, for PH.D. at University of Maryland.

Montgomery, Mabel Ph.D., SUNY at Buffalo 24 years

Stern, Samuel T. Ph.D., SUNY at Buffalo 9 years

Torchinelli, Guy Ph.D., M. S. University Illinois 13 years