User blog comment:Ynought/Another question/@comment-34422464-20190709152325/@comment-35470197-20190709152753

Wrong. The inequality \(< \omega_1^{\textrm{CK}}\) is irrelevant to the strength of first order logic. Even if you employ a stronger logic \(L\), you do not have \(\textrm{PTO}(L) \geq \omega_1^{\textrm{CK}}\).