Predicative Collapsing Principles

Abstract

We show that arithmetical transfinite recursion is equivalent to a suitable formalization of the following: For every ordinal α there exists an ordinal β such that 1 + β · (β + α) (ordinal arithmetic) admits an almost order preserving collapse into β. Arithmetical comprehension is equivalent to a statement of the same form, with β · α at the place of β · (β + α). We will also characterize the principles that any set is contained in a countable coded ω-model of arithmetical transfinite recursion resp. arithmetical comprehension.

Minimal bad sequences are necessary for a uniform Kruskal theorem

“…In [12] the order type of ϑ(D) has been determined for some natural WO-dilators D. For example, it has been shown that D(X) = 1 + 2 × X 2 (with an appropriate linear order) leads to ϑ(D) ∼ = Γ 0 . By the uniform independence principle we can now recover Friedman's result [16] that Kruskal's theorem for binary trees with two labels is independent of the axiom system ATR 0 .…”

Bachmann–Howard derivatives

“…. , z k , the order on ω 2 (Z ) is characterized by the following clauses (as before Theorem 2.2 in [24]):…”

“…For an ordinal α and a number n ∈ N, the ordinal ω α n is explained by the recursive clauses ω α 0 = α and ω α n+1 = ω ω α n (towers of exponentials in the sense of ordinal arithmetic). The resulting ordinal ω α 2 is isomorphic to the order ω 2 (1 + α) from (1.3), as shown in [24]. We now define linear orders OT 0 n by the recursive clauses…”

“…. Secondly, iterated applications of the result from [24] show that OT n [0] has order type ω 2n−1 := ω 1 2n−1 = ω 0 2n , for any n > 0. In [20] this fact was established by explicit computations that involve the addition-free Veblen functions, which may seem somewhat ad hoc.…”