June 2011 The Veblen functions for computability theorists
Alberto Marcone, Antonio Montalbán
J. Symbolic Logic 76(2): 575-602 (June 2011). DOI: 10.2178/jsl/1305810765


We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) “If 𝒳 is a well-ordering, then so is ε𝒳”, 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 ACA₀⁺ over RCA₀. To prove the latter statement we need to use ωα iterations of the Turing jump, and we show that the statement is equivalent to Π⁰ωα-CA₀. 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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Alberto Marcone. Antonio Montalbán. "The Veblen functions for computability theorists." J. Symbolic Logic 76 (2) 575 - 602, June 2011. https://doi.org/10.2178/jsl/1305810765


Published: June 2011
First available in Project Euclid: 19 May 2011

zbMATH: 1220.03050
MathSciNet: MR2830417
Digital Object Identifier: 10.2178/jsl/1305810765

Rights: Copyright © 2011 Association for Symbolic Logic


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Vol.76 • No. 2 • June 2011
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