## Journal of Symbolic Logic

### On the Consistency of the Definable Tree Property on $\aleph_1$

Amir Leshem

#### Abstract

In this paper we prove the equiconsistency of "Every $\omega_1$-tree which is first order definable over (H$_{\omega_1}\cdot\varepsilon$) has a cofinal branch" with the existence of a $\Pi^1_1$ reflecting cardinal. We also prove that the addition of MA to the definable tree property increases the consistency strength to that of a weakly compact cardinal. Finally we comment on the generalization to higher cardinals.

#### Article information

Source
J. Symbolic Logic, Volume 65, Issue 3 (2000), 1204-1214.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183746177

Mathematical Reviews number (MathSciNet)
MR1791372

Zentralblatt MATH identifier
0968.03058

JSTOR
Leshem, Amir. On the Consistency of the Definable Tree Property on $\aleph_1$. J. Symbolic Logic 65 (2000), no. 3, 1204--1214. https://projecteuclid.org/euclid.jsl/1183746177