An order-theoretic characterization of the Howard–Bachmann-hierarchy

Abstract

In this article we provide an intrinsic characterization of the famous Howard-Bachmann ordinal in terms of a natural well-partial-ordering by showing that this ordinal can be realized as a maximal order type of a class of generalized trees with respect to a homeomorphic embeddability relation. We use our calculations to draw some conclusions about some corresponding subsystems of second order arithmetic. All these subsystems deal with versions of light-face Π 1 1 -comprehension.

Proof-Theoretic Strengths of Weak Theories for Positive Inductive Definitions

“…From Proposition 1.3.2 we see that Π 0 k (Ω)-ID(Π 0 1 ) is interpreted canonically in RCA 0 + (Π 1 1 (Π 0 k+2 )-CA) − . As in [11] using [6] we see that Π 1 k+1 -BI 0 comprises RCA 0 + (Π 1 1 (Π 0 k+2 )-CA) − . In [8] it is shown that…”

“…Note that the system OT ′ (ϑ) is ω-exponential-free except ϑ(α) = ω α0 for some α 0 . An inspection of the proof in [11] shows that Acc-ID(Acc) suffices to prove the wellfoundedness of ordinals up to each ordinal< ϑ(Ω · ω). Let < be the elementary recursive relation obtained from the relation < on OT ′ (ϑ) through a suitable encoding.…”

“…We show the lemma by induction on a. First let us verify the condition (11) in Hâ +1 ⊢ ψâ 1 Γ. From γ <â + 1 we see k(Γ) ⊂ H γ ⊂ Hâ +1 .…”

Ordinal notation systems corresponding to Friedman’s linearized well-partial-orders with gap-condition

“…It is our general belief that this is a maximal linear extension. In [18,19] we already obtained partial results concerning this conjecture. In this paper, we want to investigate whether this is also true for the linearized version of the gap-embeddability relation, i.e.…”

A flexible type system for the small Veblen ordinal

“…FIT stands for "theory for function(al)s, non-iterated inductive definitions, and types (of level 1)", and it represents the first step for a generalization of the theory in [Fef92] which turns out to have the small Veblen ordinal as measure for its proof-theoretic strength, i.e., ϑΩ ω when using the terminology of [RW93]. Theories that have ϑΩ ω as proof-theoretic strength are for instance Π 1 2 -BI 0 from [RW93] or more recently RCA 0 + (Π 1 1 (Π 0 3 )-CA 0 ) − from [VRW17]. While these theories are analyzed by impredicative proof-theoretic methods, our treatment of FIT uses metapredicative methods for the lower bound.…”