## Journal of Symbolic Logic

- J. Symbolic Logic
- Volume 65, Issue 3 (2000), 1311-1320.

### Stationary Sets and Infinitary Logic

Saharon Shelah and Jouko Vaananen

#### Abstract

Let K$^0_\lambda$ be the class of structures $\langle\lambda,<, A\rangle$, where $A \subseteq \lambda$ is disjoint from a club, and let K$^1_\lambda$ be the class of structures $\langle\lambda,<,A\rangle$, where $A \subseteq \lambda$ contains a club. We prove that if $\lambda = \lambda^{<\kappa}$ is regular, then no sentence of L$_{\lambda+\kappa}$ separates K$^0_\lambda$ and K$^1_\lambda$. On the other hand, we prove that if $\lambda = \mu^+,\mu = \mu^{<\mu}$, and a forcing axiom holds (and $\aleph^L_1 = \aleph_1$ if $\mu = \aleph_0$), then there is a sentence of L$_{\lambda\lambda}$ which separates K$^0_\lambda$ and K$^1_\lambda$.

#### Article information

**Source**

J. Symbolic Logic, Volume 65, Issue 3 (2000), 1311-1320.

**Dates**

First available in Project Euclid: 6 July 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.jsl/1183746182

**Mathematical Reviews number (MathSciNet)**

MR1791377

**Zentralblatt MATH identifier**

0969.03050

**JSTOR**

links.jstor.org

#### Citation

Shelah, Saharon; Vaananen, Jouko. Stationary Sets and Infinitary Logic. J. Symbolic Logic 65 (2000), no. 3, 1311--1320. https://projecteuclid.org/euclid.jsl/1183746182