From daemon Thu Nov 17 01:49:10 1994
From: greg@cco.caltech.edu (Gregory Hjorth)
Date: Wed, 16 Nov 1994 16:40:32 -0800
To: miller@math.wisc.edu
Subject: Q10.4
%
dear professor miller,
at 10.4 on page 650 of israel conference proceedings editored
by haim judah you ask whether there are any complete theories
realised by exactly an unbounded subset of \omega_1^L. if i
understand the problem correctly, then i think it has a short
solution.
let X be the set of complete theories that satisfy "everything
is countable" and have unboundedly many \alpha<\omega_1^L
realising them. the theory of L_{\omega_1^L} is one such theory,
and we will be done if we prove that there are some others.
but X is a definable class in L_{\omega_1^L}, and so it must
have some other elements, or L_{\omega_1^L} would admit a truth
defintion (\phi iff the unique theory in X satisfies \phi).
with a bit more effort i think we can show X is uncountable
(in L).
yours, greg.