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Ordinal inequalities, transfinite induction, and reverse mathematics

Published online by Cambridge University Press:  12 March 2014

Jeffry L. Hirst*
Affiliation:
Department of Mathematical Sciences, Appalachian State University, Boone, NC 28608, E-mail: jlh@math.appstate.edu

Abstract

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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1999

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References

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