Omitting Types and the Logic Action

Gian Arturo Marco

This research is on the omitting types property for Borel and Analytic sets in Polish G spaces, where G is any subgroup of the symmetric group S_{∞} endowed with the pointwise topology.

S_{∞} and its subgroups act on the space of countable structures for a given theory in a countable language. This space is universal for the S_{∞}-action.

We give a topological characterization of the notion of omitting types as a special requirement of orbit-separation or of boundedness of the orbital rank.

We say that a countable structure M strongly omits a non isolated orbit p if no suborbit q of p is locally dense; we show that this is equivalent to the following: q is not separable from p by a q dominating subtype s such that q is splittable with, respect to s. In general we prove that omitting types fails for a countable structure M whenever we can produce a G embedding between orbits that is not in the automorphism group of M.

We show that the existence of such a G-embedding depends on the presence of a group-stabilizer that is not a Jordan group.

Let M be a countable structure in a countable language; the main statement of this paper is the following:

Theorem 1: If G ⊂ S_{∞} is closed in the Baire space ω^{ω} , G = Aut(M), and every M-type is a Jordan set, then the G omitting types holds for M.

As a corollary we have:

Corollary 2: The Scott sentence of M has no model of height ω, 1-elementary equivalent to M.

Address: C.so S. Martino n’8, TURIN ITALY

E-mail: giannimarco@tin.it