# [FOM] Borel categoricity

Dave Marker marker at math.uic.edu
Thu Jul 10 08:43:44 EDT 2003

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Steve's posting reminds me on an interesting question that I believe
is still open.

Is Morley's categoricity theorem true for Borel models? (i.e.
if  T is a countable theory that is not aleph_1-categorical
are there nonisomorphic Borel models?) One could also consider
Borel isomorphism, but I expect that question would have a very
different flavor.

One might hope to answer this positivly using the Baldwin-Lachlan
characterization that a countable  theory is uncountably categorical if
and only if it is omega-stable and has no Vaughtian pairs.

Let T be a complete theory in a countable language. You need to know
4 things.

i) If there are uncountably many types over some countable set
then there is a Borel model that realizes uncountably many types over
some countable subset.

ii) There is a Borel model that realizes only countably many types
over every countable subset.

iii) For any formula phi defining an infinite set there is a Borel
model where uncountably many things satisfy phi.

iv) If T has a two cardinal model then T has a Borel two cardinal model.
(if you've done i) and ii) it might be helpful to assume omega-stability
here)

When doing this for arbitrary models i) and iii) are easy.
ii) was proved by Morley using a well ordered set of indiscernibles
(Thm 5.2.9 in my model theory book) and iv) is a consequence of
Vaught's two cardinal theorem (Thm 4.3.34 and Cor 4.3.39 of my book).
Both proofs make heavy use of uncountable well orders.

In the Borel context iii) follows easily from Harvey's proof of the
Borel completeness theorem.

Woodin proved i) using forcing. Knight and Simpson gave
a combinatorial proof. This is a consequence of Schmerl's theorem
as well.

As far as I know ii) and iv) are still open.

Dave

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