 Gentzen’s consistency proof for PA employs a reduction procedure \(\cR\) on proofs P of the empty sequent together with an assignment ord of representations for ordinals to proofs such that \(\ord(\cR(P))&lt; \ord(P)\). Here \(&lt;\) denotes the ordering on ordinal representations induced by the ordering of the pertinent ordinals. For this purpose he needed representations for ordinals \(&lt;\varepsilon_0\) where \(\varepsilon_0\) is the smallest ordinal \(\tau\) such that whenever \(\alpha&lt;\tau\) then also \(\omega^{\alpha}&lt;\tau\) with \(\alpha\mapsto \omega^{\alpha}\) being the function of ordinal exponentiation with base \(\omega\). Moreover, the functions \(\cR\) and ord and the relation \(&lt;\) are computable (when viewed as acting on codes for the syntactic objects), they can be chosen to be primitive recursive. With \(g(n)= \ord(\cR^n(P))\), the n-fold iteration of \(\cR\) applied to P, one has \(g(0)&gt;g(1)&gt; g(2)&gt; \ldots &gt; g(n)\) for all n, which is absurd as the ordinals \(&lt;\varepsilon_0\) are well-founded. Hence PA is consistent.