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 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 Πωα -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.