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

TL;DR

This paper offers an intrinsic order-theoretic characterization of the Howard-Bachmann ordinal using well-partial-orderings and generalized trees, linking it to subsystems of second-order arithmetic with $\Pi^1_1$-comprehension.

Contribution

It introduces a novel intrinsic characterization of the Howard-Bachmann ordinal via well-partial-orderings and generalized trees, connecting ordinal analysis with subsystems of second-order arithmetic.

Findings

Howard-Bachmann ordinal characterized as maximal order type

Connection established between order types and $\Pi^1_1$-comprehension subsystems

Insights into the structure of light-face $\Pi^1_1$-comprehension systems

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 $\Pi^1_1$-comprehension