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The Veblen functions for computability theorists

Published online by Cambridge University Press:  12 March 2014

Alberto Marcone  and
Antonio Montalbán
Alberto Marcone
Affiliation:
Dipartimento di Matematica e Informatica, Università di Udine33100 Udine, Italy, E-mail: alberto.marcone@dimi.uniud.it
Antonio Montalbán
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Il 60637, USA, E-mail: antonio@math.uchicago.edu
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Abstract

We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) “If is a well-ordering, then so is ”, and (2) “If is a well-ordering, then so is φ(α, )”, where ∝ 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 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 . Our proofs are purely computability-theoretic. We also give a new proof of a result of Friedman: the statement “if is a well-ordering, then so is φ(, 0)” is equivalent to ATR0 over RCA0.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 2 , June 2011 , pp. 575 - 602
Copyright
Copyright © Association for Symbolic Logic 2011

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