Published online by Cambridge University Press: 12 March 2014
If α and β are ordinals, α ≤ β, and β ≰ α then α+ 1 < β. The first result of this paper shows that the restriction of this statement to countable well orderings is provably equivalent to ACA0, 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 ACA0 and an arithmetical transfinite induction scheme.
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