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Faculty Feature
Professor Jean E. Rubin has been a member of the Mathematics faculty at Purdue since 1967. After receiving her Ph.D. at Stanford, she held positions at the University of Oregon and Michigan State before coming to West Lafayette. Her research has focused primarily on logic and set theory. Professor Rubin has published many research papers and is the author of five books, which she describes below.
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- Equivalents of the Axiom of Choice (with H. Rubin), North Holland, 1963 (reprinted 1970).
- Set Theory for the Mathematician, Holden-Day, 1967.
- Equivalents of the Axiom of Choice II (with H. Rubin), North Holland, 1985.
- Mathematical Logic: Applications and Theory, Saunders, 1990.
- Consequences of the Axiom of Choice (with P. Howard), accepted for publication, June 1997, American Mathematical Society.
[2] is a text for a graduate course in set theory, and [4] is a text for an undergraduate course in logic. The others are research monographs.
Much of my research deals with a non-constructive principle called the Axiom of Choice. In one of its equivalent forms, it asserts that for every set S of non-empty sets, there is a function f which chooses an element from each set in S. The Axiom of Choice is not terribly earth-shaking, and most mathematicians use it in their work without any qualms. It is, however, a non-constructive principle (there is no rule given for constructing the function f), and some of its consequences are rather unusual. For example, the following statement, called the Banach-Tarski paradox, is a consequence of the Axiom of Choice:
Given any sphere, S, in 3-dimensions, it is possible to subdivide it into a finite number of parts, move these parts using rigid motion (no stretching or contracting), then put them together again into a sphere that is one half the size of the original sphere, S. (The phrase "one half the size" could be replaced by "three times the size," "one tenth the size," etc.)
In [1] and [3], Herman (Herman Rubin, Professor of Statistics at Purdue) and I have collected a large number of statements in all areas of mathematics that are equivalent to the Axiom of Choice. In our latest work [5], Paul Howard (Professor of Mathematics at Eastern Michigan University) and I study hundreds of non-constructive statements, again in all areas of mathematics, which are consequences of the Axiom of Choice.
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