User blog comment:Ynought/Another question/@comment-34422464-20190709152325/@comment-35470197-20190709154134

No, no. PTO is defined as the least recursive ordinal \(\alpha\) such that for any recursive well-ordering whose ordinal type is \(\alpha\), its well-foundedness is unprovable. Therefore it never reaches \(\omega_1^{\textrm{CK}}\). If you consider a sufficiently strong non-recursive theory, then PTO just becomes ill-defined.