Cofinalities of elementary substructures of structures on aleph_omega
Logic
2016-09-06 v1
Abstract
Let 0<n^*< omega and f:X-> 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.
Cite
@article{arxiv.math/9604242,
title = {Cofinalities of elementary substructures of structures on aleph_omega},
author = {Kecheng Liu and Saharon Shelah},
journal= {arXiv preprint arXiv:math/9604242},
year = {2016}
}