Ordinal inequalities, transfinite induction, and reverse mathematics

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

If α and β are ordinals, α ≤ β, and $\beta \nleq \alpha$ , 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 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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References found in this work

Subsystems of Second Order Arithmetic.Stephen G. Simpson - 1999 - Studia Logica 77 (1):129-129.
Reverse mathematics and ordinal exponentiation.Jeffry L. Hirst - 1994 - Annals of Pure and Applied Logic 66 (1):1-18.

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