Nonetheless, Parikh showed that for appropriate choices of \(\tau\), any proof of a contradiction in \(\mathsf{PA}^F\) must itself be very long. For instance, if we consider the super-exponential function \(2 \Uparrow 0 = 1\) and \(2 \Uparrow (x+1) = 2^{2 \Uparrow x}\) and let \(\tau\) be the primitive recursive term \(2 \Uparrow 2^k\), it is a consequence of Parikh’s result that any proof of a contradiction in \(\mathsf{PA}^F\) must be on the order of \(2^k\) steps long. e. non-vague) property