  Hilbert never gave a general account of which operations and methods of proof are acceptable from the finitist standpoint, but only examples of operations and methods of inference in contentual finitary number theory which he accepted as finitary. Contentual induction was accepted in its application to finitary statements of the hypothetical, general kind explicitly in  Hilbert (1922b).  He  (1923,  1139) said that intuitive thought “includes recursion and intuitive induction for finite existing totalities,” and used exponentiation in an example in  1928.  Bernays (1930) explained how exponentiation may be understood as a finitary operation on numerals.  Hilbert and Bernays (1934)  give the only general account of finitary contentual number theory; according to it, operations defined by primitive recursion and proofs using induction are finitarily acceptable. All of these methods can be formalized in a system known as primitive recursive arithmetic (\(\PRA\)), which allows definitions of functions by primitive recursion and induction on quantifier-free formulas (ibid.). However, neither Hilbert nor Bernays ever claimed that only primitive recursive operations count as finitary, and they in fact did use some non-primitive recursive methods in ostensibly finitary consistency proofs already in 1923 (see  Tait 2002  and  Zach 2003).