Slow consistency (Q1935874)

The authors introduce the notion of slow consistency \(\mathrm{Con}^*(\mathrm{PA})\) of Peano arithmetic \(\mathrm{PA}\): ``For all \(x\), the theory consisting of the first \(f(x)\) non-logical axioms of \(\mathrm{PA}\) is consistent'', where \(f\) is a certain slow-growing computable function. Then the authors prove that the strength of the theory \(\mathrm{PA} + \mathrm{Con}^*(\mathrm{PA})\) is strictly between the strength of the theory \(\mathrm{PA}\) and the strength of the theory \(\mathrm{PA} + \mathrm{Con}(\mathrm{PA})\).