Ordinal Inequalities, Transfinite Induction, and Reverse Mathematics

Journal of Symbolic Logic 64 (2):769-774 (1999)
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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.



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