## Journal of Symbolic Logic

### Ordinal Inequalities, Transfinite Induction, and Reverse Mathematics

Jeffry L. Hirst

#### Abstract

If $\alpha$ and $\beta$ are ordinals, $\alpha \leq \beta$, and $\beta \nleq \alpha$, then $\alpha + 1 \leq \beta$. The first result of this paper shows that the restriction of this statement to countable well orderings is provably equivalent to ACA$_0$, a subsystem of second order arithmetic introduced by Friedman. The proof of the equivalence is reminiscent of Dekker's construction of a hypersimple set. An application of the theorem yields the equivalence of the set comprehension scheme ACA$_0$ and an arithmetical transfinite induction scheme.

#### Article information

Source
J. Symbolic Logic, Volume 64, Issue 2 (1999), 769-774.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745808

Mathematical Reviews number (MathSciNet)
MR1777785

Zentralblatt MATH identifier
0930.03085

JSTOR