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 ACA + 0 over RCA 0 . 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 Π 0 ω α -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 ϕ(X , 0)" is equivalent to ATR 0 over RCA 0 ., Z(12), · · · Proof. Suppose first that Y = J (Z). When n = 0 let σ n = Z ↾ 1, which works by (P1). When n > 0 let. This is the base step of an induction that, using the same argument, shows that Y (i) = J(σ n )(i) for every i < n. Thus Y ↾ n ⊆ J(σ n ). By (P6), we have Y ↾ n ∈ J(N <N ) and we can apply (P3) and (P5) to obtain Y ↾ n = J(σ n ) and |σ n | > n. Now assume that (2) holds, and suppose towards a contradiction that Y = J (Z). Let n be least such that Y (n − 1) = J (Z)(n − 1). If σ n ⊂ Z is such that Y ↾ n = J(σ n ) we have J(σ n )(n−1) = J (Z)(n−1). This can occur only if {n−1} σn (n−1)↑ and {n − 1} Z (n − 1)↓, which implies n ′ > |σ n |, where n ′ = |J (Z)(n − 1)|. Notice that for any m > n ′ we have J(Z ↾ m)(n − 1) = J (Z)(n − 1) and hence J(Z ↾ m)(n − 1) = Y (n − 1). This contradicts the existence of σ n ′ ⊂ Z with |σ n ′ | > n ′ such that Y ↾ n ′ = J(σ n ′ ).The following corollary is obtained by iterating the Lemma.Corollary 4.6. For every m > 0, given Y, Z ∈ N N , the following are equivalent:(1) Y = J m (Z);(2) for every n there exists σ n ⊂ Z with |σ n | ≥ n+m such that Y ↾ n = J m (σ n ).The Jump function leads to the definition of the Jump Tree.Definition 4.7. Given a tree T ⊆ N <N we define the Jump Tree of T to beThe following lemmas summarize the main properties of the Jump Tree. Lemma 4.8. For every tree T , J T (T ) is a tree computable in T . Proof. J T (T ) is a tree because if τ ⊂ J(σ) for σ ∈ T , then τ = J(K(τ )) (by (P6) and (P3)) and K(τ ) ∈ T (since by (P6), (P2) and (P1), K(τ ) ⊂ K(J(σ)) ⊆ σ). J T (T ) is computable in T because τ ∈ J T (T ) if and only if τ = J(K(τ )) (which is equivalent to τ ∈ J(N <N ) by (P3)) and K(τ ) ∈ T .Lemma 4.9. For every tree T , [J T (T )] = { J (Z) : Z ∈ [T ] }.Definition 5.4. We define the ω-Jump operator to be the function J ω :We need to show that the union above makes sense. Assume Y = J ω (Z). It might help to look at Figure 1. Notice that for each n, Y (n) ⊂ J n (Z) because for every X, J (X)(0) ⊂ X and Y (n) = J (J n (Z))(0). We also know that if σ ⊂ J (X), then K(σ) ⊂ X, so that K n (Y (n)) ⊂ Z. It follows that n K n (Y (n)) ⊆ Z. Applying (P5) n times we get that |K n (Y (n))| > n, and therefore the union above does actually produce Z. We have just proved the fol...

From hierarchies to well-foundedness

“…Note however, that the theory presented in [6] is parameter free. Also note that the references use slightly different notations; Π 0 α -CA 0 in [5] and ACA − α in [6, §4].…”

Reverse Mathematics and Well-ordering Principles

“…Recently two new results appeared in preprints [10,7]. These result give characterizations of the form (1) for the theories ACA + 0 and ATR 0 , respectively, in the form of familiar proof-theoretic functions.…”

Predicative Collapsing Principles

“…The statement that ε X is well-founded for any well-order X is equivalent to the assertion that the ω-jump of any set exists, as shown by A. Marcone and A. Montalbán [14] (see also the proof-theoretic argument due to B. Afshari and M. Rathen [3]). Together with Theorem 2.6 we obtain the following:…”

“…Precise definitions and proofs can be found in the references that are given in the right column. 13] the ω-jump of every set exists X → ε X [14,3] arithmetical transfinite recursion…”