Computing Maximal Chains

TL;DR

The paper proves that while maximal chains exist in all well partial orders, they cannot be computed by hyperarithmetic sets, but almost all sets in the sense of category can compute such chains.

Contribution

It establishes the non-computability of maximal chains in all computable wpos by hyperarithmetic sets and characterizes the sets capable of computing strongly maximal chains.

Findings

Hyperarithmetic sets cannot compute maximal chains in all computable wpos.

Almost every set in the sense of category can compute maximal chains.

Sets that compute strongly maximal chains are exactly those that compute all hyperarithmetic sets.

Abstract

In 1967 Wolk proved that every well partial order (wpo) has a maximal chain; that is a chain of maximal order type. (Note that all chains in a wpo are well-ordered.) We prove that such maximal chain cannot be found computably, not even hyperarithmetically: No hyperarithmetic set can compute maximal chains in all computable wpos. However, we prove that almost every set, in the sense of category, can compute maximal chains in all computable wpos. Wolk's original result actually shows that every wpo has a strongly maximal chain, which we define below. We show that a set computes strongly maximal chains in all computable wpo if and only if it computes all hyperarithmetic sets.