Suppose that R is a countable relation on the Turing degrees. Then R can be defined in D, the Turing degrees with â‰¤T , by a first order formula with finitely many parameters. The parameters are builtâ€¦ (More)

Is this really evidence (as is often cited) that the Continuum Hypothesis has no answer? Another prominent problem from the early 20th century concerns the projective sets, [8]; these are the subsetsâ€¦ (More)

Proceedings of the National Academy of Sciencesâ€¦

1983

It is shown that, within L(R), the smallest inner model of set theory containing the reals, the axiom of determinacy is equivalent to the existence of arbitrarily large cardinals below Theta with theâ€¦ (More)

We prove that every countable relation on the enumeration degrees, E, is uniformly definable from parameters in E. Consequently, the first order theory of E is recursively isomorphic to the secondâ€¦ (More)

After small forcing, almost every strongness embedding is the lift of a strongness embedding in the ground model. Consequently, small forcing creates neither strong nor Woodin cardinals. The widelyâ€¦ (More)

The usual definition of the set of constructible reals R is Î£2. This set can have a simpler definition if, for example, it is countable or if every real is in L. Martin and Solovay [MS1] have shownâ€¦ (More)

The Ground Axiom asserts that the universe is not a nontrivial set-forcing extension of any inner model. Despite the apparent second-order nature of this assertion, it is first-order expressible inâ€¦ (More)

Proceedings of the National Academy of Sciencesâ€¦

1988

It is shown that if there exists a supercompact cardinal then every set of reals, which is an element of L(R), is the projection of a weakly homogeneous tree. As a consequence of this theorem andâ€¦ (More)

This problem belongs to an ever-increasing list of problems known to be unsolvable from the (usual) axioms of set theory. However, some of these problems have now been solved. But what does thisâ€¦ (More)