2012 seminar talk: Isomorphism of Computable Structures and Vaught's Conjecture
Talk held by Howard Becker (Universität Münster, Germany)
at the KGRC seminar on 2012-01-12.
Abstract
The following question is open: Does there exist a
hyperarithmetic class of computable structures with at least one, but only
finitely many, properly $\Sigma^1_1$ isomorphism classes? Given any
oracle $x$ in $2^\omega$, we can ask the same question relativized to $x$
(that is, replace hyperarithmetic, computable, and $\Sigma^1_1$ by
hyperarithmetic-in-$x$, computable-in-$x$, and $\Sigma^1_1$-in-$x$). A
negative answer for all $x$ implies Vaught's Conjecture for infinitary
logic.