Computing Maximal Chains
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 wellordered.) 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2012
 DOI:
 10.48550/arXiv.1201.4408
 arXiv:
 arXiv:1201.4408
 Bibcode:
 2012arXiv1201.4408M
 Keywords:

 Mathematics  Logic;
 Mathematics  Combinatorics;
 03D80 (Primary) 06A07 (Secondary)
 EPrint:
 Archive for Mathematical Logic, 51 (2012), 651660