  As mentioned above, initially Gödel and Bernays thought that the difficulty for the consistency proof of \(\PA\) could be overcome by employing methods which, although not formalizable in \(\PA\), are nonetheless finitary. Whether such methods would be considered finitary according to the original conception of finitism or constitute an extension of the original finitist viewpoint is a matter of debate. The new methods considered included a finitary version of the \(\omega\)-rule proposed by Hilbert  (1931b;  1931a).  It is fair to say, however, that after about 1934 it has been almost universally accepted that the methods of proof accepted as finitary prior to Gödel’s results are all formalizable in \(\PA\). Extensions of the original finitist standpoint have been proposed and defended on broadly finitary grounds, e.g.,  Gentzen (1936)  defended the use of transfinite induction up to \(\varepsilon_0\) in his consistency proof for \(\PA\) as “indisputable,”  Takeuti (1987)  gave another defense.  Gödel (1958)  presented another extension of the finitist standpoint; the work of Kreisel mentioned above may be seen as another attempt to extend finitism while retaining the spirit of Hilbert’s original conception.