Some Results on Stationary Subsets of Regular Cardinals

Mika Rautila

We will discuss of some new results on stationary subsets of regular cardinals. A result establishes a condition in terms of cub-games when a stationary subsets can be destroyed without adding subsets of small cardinality.

As a model theoretic application of the results we have the following theorem:

Theorem 1 Let T be a countable complete unsuperstable theory and κ a cardinal. Suppose that one of the following hold:

1. The cardinal κ is singular or κ = ω or there is a stationary set S ⊆ S_{ω}^{κ^{+}} with □κS.

2. The cardinal κ is regular, κ^{ω} = κ and the ideal I[κ^{+}] is improper.

Then there are models 𝔄, 𝔅 ⊧ T of cardinality κ^{+} such that 𝔄 ≅ 𝔅̸ and a partial order P that does not add subsets of cardinality at most κ such that in every P -generic extension 𝔄 ≅ 𝔅.

Address: University of Helsinki, PL 4, 00014 University of Helsinki

E-mail: mika.rautila@helsinki.fi