# [FOM] VAught's conj and Lowenheim Skolem in infinitary logic

John Baldwin jbaldwin at uic.edu
Mon Feb 27 22:37:58 EST 2006

```For a first order theory T with no finite models, categoricity in
some cardinal kappa of T (all models of T with cardinality kappa
are isomorphic) implies T is complete.

In the absence of upward Lowenheim Skolem, the analog for L omega
_1 omega fails.

By the downward Lowenheim Skolem theorem, every sentence  sigma of
L omega _1 omega has countable `elementary submodel' - satisfies
the same sentences in the fragment generated by sigma.  But it is
easy to construct models which have no countable submodel
satisfying all the same sentences of  L omega _1 omega.

Nevertheless a `reduction' of studying categorical sentences to
complete sentences is an essential step in the study of
categorical sentences in L omega _1 omega.

Since the sentence has a countable model, each of its models
realizes only countably many types over the empty set.

The key step (Shelah) is to show if a sentence sigma is
categorical in some uncountable cardinal kappa then there is a
sigma' which implies sigma, has a model of power kappa, and is
complete.

This is done by a nice argument using the Lopez-Escobar theorem on
non-definability of linear order.

As an easy  consequence of the argument I showed that if there is
a counterexample sigma to Vaught's conjecture there is one which
realizes only countably many types over the empty set.  (This was
shown much earlier by Makkai, using reasonably serious apparatus
of admissible model theory, including Ressayre's notion of a
saturated model for an admissible fragment.)

By an clever combination of model theoretic and descriptive set
theoretic methods, Hjorth has shown that if there is a
counterexample to Vaught's conjecture there is one which has no
model of cardinality aleph_2.

I observed that any first order counterexample to Vaught's
conjecture must have 2^aleph_1 models of cardinality aleph_1.
(This is trivial from two hard theorems of Shelah.  For a sentence
of  L omega _1 omega it is open.  The first step in attempting
such an extension by a parallel proof would be to show that a
sentence of L omega _1 omega has few models in aleph_1, it is
omega stable; this is known to be independent of ZFC.

references: see Hjorth's webpage:
http://www.math.ucla.edu/~greg/research.html

the articles on Knight's model... (the Knight is Julia) and `a
note on counterexamples to Vaught's conjecture'

the Shelah archive: 48, 87a,87b, 88  (for the independence from
ZFC)   http://shelah.logic.at/

or my webpage:  http://www2.math.uic.edu/~jbaldwin/model.html

chapter 7 of the monograph for the completeness issues; the paper
on Vaught's conjecture

John T. Baldwin
Director, Office of Mathematics Education
Department of Mathematics, Statistics,
and Computer Science  M/C 249
jbaldwin at uic.edu
312-413-2149
Room 327 Science and Engineering Offices (SEO)
851 S. Morgan
Chicago, IL 60607

Assistant to the director
Jan Nekola: 312-413-3750
```

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