Dave Marker marker at math.uic.edu
Mon Jun 30 12:34:06 EDT 2003

While categoricity notions are defined in terms of infinite cardinals
and well orders, they have much simpler characterizations. Let T be a
complete theory in a countable language.

*(Ryll-Nardewski) T is countably categorical if and only if
for all n there are only finitely many inequivalent formulas
in n-free variables.

Thus countable categoricity is an arithmetic condition about T.

* (Baldwin-Lachlan [after Morley]) T is categorical in some uncountable
cardinal if and only if T is uncountable in every uncountable power
if and only if a) T is omega-stable and
b) T has no Vaughtian pairs.

a) and b) only refer to the countable models of T.
omega-stability is a Pi^1_1 notion, while "no Vaughtian pairs"
is arithmetic.

My recollection is that all of these characterizations are "best
possible" in that if you look at the descriptive set theoretic
complexity of the characterization, they are complete in that class.
(for example omega-stability is Pi^1_1-complete, as is superstability,
but stability is \Pi^0_2-complete).

As far as I know most of this is just folklore, but was certainly known by
Gerald Sacks and the people around him in the early 70s.

Dave Marker

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