The Veblen functions for computability theorists

Abstract

We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) “If 𝒳 is a well-ordering, then so is ε_𝒳”, 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 ω^α iterations of the Turing jump, and we show that the statement is equivalent to Π⁰_ω^α-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.