## Journal of Symbolic Logic

### Stationary Subsets of $\lbrack \aleph_\omega \rbrack^{<\omega_n}$

Kecheng Liu

#### Abstract

In this paper, assuming large cardinals, we prove the consistency of the following: Let $n \in \omega$ and $k_1, k_2 \leq n$. Let $f: \omega \rightarrow \{k_1, k_2\}$ be such that for all $n_1 < n_2 \in f^{-1}\{k_1\}, n_2 - n_1 \geq 4$. Then the set $S = \{x \subset \aleph_\omega:|x| = \omega_n \text{and} \forall m \geq n, cf(x \cap \omega_m) = \omega_{f(m)}\}$ is stationary in $\lbrack \aleph_\omega \rbrack^{<\omega_{n + 1}}$. The above is equivalent to the statement that for any structure $\mathscr{A}$ on $\aleph_\omega$, there is $\mathscr{B} \prec A$ such that $|\mathscr{B}| = \omega_n$ and for all $m > n, cf(\mathscr{B} \cap \omega_m) = \omega_{f(m)}$.

#### Article information

Source
J. Symbolic Logic, Volume 58, Issue 4 (1993), 1201-1218.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183744371

Mathematical Reviews number (MathSciNet)
MR1253918

Zentralblatt MATH identifier
0794.03069

JSTOR
Liu, Kecheng. Stationary Subsets of $\lbrack \aleph_\omega \rbrack^{&lt;\omega_n}$. J. Symbolic Logic 58 (1993), no. 4, 1201--1218. https://projecteuclid.org/euclid.jsl/1183744371