Marion Scheepers

Set Theory and its Relatives


Set Theory, a foundation for mathematics, has numerous sub-disciplines and emphases. It draws heavily on mathematical logic and extends several classical lines of investigation related to computability, computational complexity theory and combinatorics to the transfinite. As mathematics has evolved over the centuries many mathematical statements about the objects of study have been formulated as questions or as conjectures. In light of now classical work of Godel it is known that not all statements true about certain structures have mathematical proofs. This fact raises the possibility that mathematical questions or conjectures might not be answerable on the basis of the standard axioms of Mathematics, using the mathematical method. Set Theory develops techniques for determining whether questions or conjectures have this undecidability feature relative to a given axiom system and for measuring the relative strength of mathematical statements whose consistency or independence cannot be established by merely postulating that the Zermelo-Fraenkel axioms for Mathematics are consistent, and techniques for establishing a complexity hierarchy among fundamental infinitary mathematical decision or search problems.

An outgrowth of this foundational activity is the formulation of powerful mathematical hypotheses whose relative consistency status is known, and which have rich mathematical consequences. Numerous such hypotheses are now in use in one of the usual laboratories for refining the products of Set Theory - General Topology. Analysis, Algebra and Discrete Mathematics are fast becoming standard testing grounds for the products of Set Theory.

Several products of Set Theory ought to be part of the general knowledge of all mathematicians. These include ordinal arithmetic, parts of cardinal arithmetic, hypotheses such as the Continuum Hypothesis or Martin's Axiom, the notions of a measurable cardinal or a real-valued measurable cardinal, constructibility, forcing and elementary submodels. The most specialized among these for the general mathematical audience may be Martin's Axiom, constructibility, forcing and elementary submodels. These few items constitute a set of mathematical tools that are in several subdisciplines of mathematics already as indispensable as mathematical induction, but like computability and computational complexity theory, are not treated as a necessary part of undergraduate or graduate level education in mathematics.

Here is s sample of some mathematical statements I have worked on, often with collaborators:
1. The Borel Conjecture:
In a 1919 paper in which E. Borel explored Lebesgue's notion of measurability he introduced a notion that is nowadays called strong measure zero.

2. The Bolzano Weierstrass Theorem: This classical theorem states that a bounded sequence of real numbers has a convergent subsequence.

3. Ramsey's Theorem: The infinitary version of the classical Ramsey theorem states that if the n-element subsets of an infinite set are each colored with one of a fixed finite number of colors, then some infinite set will have all its n-element subsets colored by the same color.