 At first glance Turing’s theorem seems to provide some insight into the nature of true \(\Pi^0_1\)-statements. That this is an “illusion” is revealed by the analysis of its simple proof which is just based on the trick of coding the truth of F as a member of \({\cO}\). The proof also shows that the infinitely many iterated consistency axioms \(\Con (\bT_0),\Con (\bT_1),\ldots\) of \(\bT_{{{\rsuc}}({\rlim}(e))}\) are irrelevant for proving F. As it turns out, the reason why one has to go to stage \(\omega+1\) is simply that only at stage \(\omega\) a non-standard definition of the axioms of \(\bigcup_{n&lt;\omega}\bT_n\) can be introduced. More details and other results on recursive progressions are discussed in  Appendix B.  Here let us just mention that one has considered other progressions based on various extension procedures \(\bT \mapsto \bT'\) that strengthen a given theory,  notably:[22]