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.
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