## Journal of Symbolic Logic

### Ordinal Diagrams for $\Pi_3$-Reflection

Toshiyasu Arai

#### Abstract

In this paper we introduce a recursive notation system O($\Pi_3$) of ordinals. An element of the notation system is called an ordinal diagram. The system is designed for proof theoretic study of theories of $\Pi_3$-reflection. We show that for each $\alpha < \Omega$ in O($\Pi_3$) a set theory KP $\Pi_3$ for $\Pi_3$-reflection proves that the initial segment of O($\Pi_3$) determined by $\alpha$ is a well ordering. Proof theoretic study for such theories will be reported in [4].

#### Article information

Source
J. Symbolic Logic, Volume 65, Issue 3 (2000), 1375-1394.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183746186

Mathematical Reviews number (MathSciNet)
MR1791381

Zentralblatt MATH identifier
0979.03043

JSTOR

#### Citation

Arai, Toshiyasu. Ordinal Diagrams for $\Pi_3$-Reflection. J. Symbolic Logic 65 (2000), no. 3, 1375--1394. https://projecteuclid.org/euclid.jsl/1183746186