Applications of Recursion Theoretic Methods in Set Theory

			  Theodore A. Slaman
		The University of California, Berkeley

We will discuss a collection of recent results concerning the
following question of Sierpinski, and end with some open questions.

QUESTION (Sierpinski 1936) Does there exist a set U such that every
uncountable analytic set is a continuous injective image of U.

Making nontrivial use of the Axiom of Choice, Slaman proved the following. 

THEOREM (Slaman 1998) 
   - No analytic set has the property required by Sierpinski.
   - There is a set U as required by Sierpinski.

Hjorth eliminated the use of the Axiom of Choice.

THEOREM (Hjorth 1998)  There is a co-analytic set U as required by Sierpinski.

THEOREM (Harrington 1999) The following statements are equivalent.
   - There an analytic set U such that every non-Borel analytic set is a 
     continuous injective image of U.
   - For every real x, the sharp of x exists.

We will end with a discussion of open problems and directions for
further research.