Cofinalities of elementary substructures of structures on aleph_omega

Details Cofinalities of elementary substructures of structures on aleph_omega Cofinalities of elementary substructures of structures on aleph_omega

1996-04-15

arxiv.org/abs/math/9604242v1

Israel J. Math. 99 (1997), 189-205

Mathematics

Logic

Scientific paper

Let 0 n^*+1 be a function where X subseteq omega backslash (n^*+1) is infinite. Consider the following set S_f= {x subset aleph_omega : |x| <= aleph_{n^*} & (for all n in X)cf(x cap alpha_n)= aleph_{f(n)}}. The question, first posed by Baumgartner, is whether S_f is stationary in [alpha_omega]^{< aleph_{n^*+1}}. By a standard result, the above question can also be rephrased as certain transfer property. Namely, S_f is stationary iff for any structure A=< aleph_omega, ... > there's a B prec A such that |B|= aleph_{n^*} and for all n in X we have cf(B cap aleph_n)= aleph_{f(n)}. In this paper, we are going to prove a few results concerning the above question.

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