The Veblen functions for computability theorists

We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) "If X is a well-ordering, then so is \epsilon_X", and (2) "If X is a well-ordering, then so is \phi(\alpha, X)", where \alpha is a fixed computable ordinal and \phi represents the two-placed Veblen function. For the former statement, we show that \omega iterations of the Turing jump are necessary in the proof and that the statement is equivalent to ACA_0^+ over RCA_0. To prove the latter statement we need to use \omega^\alpha iterations of the Turing jump, and we show that the statement is equivalent to \Pi^0_{\omega^\alpha}-CA_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 \phi (X,0)" is equivalent to ATR_0 over RCA_0.