7 research outputs found
On the complexity of the relations of isomorphism and bi-embeddability
Given an L_{\omega_1 \omega}-elementary class C, that is the collection of
the countable models of some L_{\omega_1 \omega}-sentence, denote by \cong_C
and \equiv_C the analytic equivalence relations of, respectively, isomorphism
and bi-embeddability on C. Generalizing some questions of Louveau and Rosendal
[LR05], in [FMR09] it was proposed the problem of determining which pairs of
analytic equivalence relations (E,F) can be realized (up to Borel
bireducibility) as pairs of the form (\cong_C,\equiv_C), C some L_{\omega_1
\omega}-elementary class (together with a partial answer for some specific
cases). Here we will provide an almost complete solution to such problem: under
very mild conditions on E and F, it is always possible to find such an
L_{\omega_1 \omega}-elementary class C.Comment: 15 page
Classes of structures with no intermediate isomorphism problems
We say that a theory is intermediate under effective reducibility if the
isomorphism problems among its computable models is neither hyperarithmetic nor
on top under effective reducibility. We prove that if an infinitary sentence
is uniformly effectively dense, a property we define in the paper, then no
extension of it is intermediate, at least when relativized to every oracle on a
cone. As an application we show that no infinitary sentence whose models are
all linear orderings is intermediate under effective reducibility relative to
every oracle on a cone
The -Vaught's Conjecture
We introduce the -Vaught's conjecture, a strengthening of the
infinitary Vaught's conjecture. We believe that if one were to prove the
infinitary Vaught's conjecture in a structural way without using techniques
from higher recursion theory, then the proof would probably be a proof of the
-Vaught's conjecture. We show the existence of an equivalent condition
to the -Vaught's conjecture and use this tool to show that all
infinitary sentences whose models are linear orders satisfy the
-Vaught's conjecture