 These proof theoretic considerations are striking and important as they involve for the first time genuine transformations of formal derivations. Nevertheless, they are preliminary as they concern a quantifier-free theory that is a part of finitist mathematics and need not be secured by a consistency proof. What has to be secured is “transfinite logic” with its “ideal elements”, as Hilbert put it later. The strategy was direct and started to emerge already in 1921. First, introduce functional terms by the transfinite  axiom[5] 