The veblen functions for computability theorists

Journal of Symbolic Logic 76 (2):575 - 602 (2011)
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Abstract

We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) "If X is a well-ordering, then so is ε X ", and (2) "If X is a well-ordering, then so is φ(α, X)", where α is a fixed computable ordinal and φ represents the two-placed Veblen function. For the former statement, we show that ω iterations of the Turing jump are necessary in the proof and that the statement is equivalent to ${\mathrm{A}\mathrm{C}\mathrm{A}}_{0}^{+}$ over RCA₀. To prove the latter statement we need to use ω α iterations of the Turing jump, and we show that the statement is equivalent to ${\mathrm{\Pi }}_{{\mathrm{\omega }}^{\mathrm{\alpha }}}^{0}{-\mathrm{C}\mathrm{A}}_{0}$ . Our proofs are purely computability-theoretic. We also give a new proof of a result of Friedman: the statement "if x is a well-ordering, then so is φ(x, 0)" is equivalent to ATR₀ over RCA₀

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Citations of this work

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Reverse mathematics: the playground of logic.Richard A. Shore - 2010 - Bulletin of Symbolic Logic 16 (3):378-402.
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References found in this work

Proof Theory and Logical Complexity.Helmut Pfeifer & Jean-Yves Girard - 1989 - Journal of Symbolic Logic 54 (4):1493.
Reverse mathematics and ordinal exponentiation.Jeffry L. Hirst - 1994 - Annals of Pure and Applied Logic 66 (1):1-18.

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