Hilbert's Finitism and the Notion of Infinity
KARL-GEORG NIEBERGALL AND MATTHIAS SCHIRN
An ihren Früchten sollt ihr sie erkennen,
heiβt es auch für die Theorien.
(By their fruits ye shall know them--
that applies also to theories.)
Hilbert, "'Über das Unendliche'", 98 
When Hilbert set about developing his foundational programme in the 1920s, proof theory was still enjoying a state of pre-Gödelian innocence. The principal objective of the programme was to establish once and for all the entire heritage of classical mathematics by means of a finitist metamathematical consistency proof. Hilbert thus aimed at justifying, by using only finitist methods, what he took to be the disputable non-finitist modes____________________
The idea of writing a paper on Hilbert's finitism grew out of our joint seminar on his prooftheoretic programme at the University of Munich in the winter term 1993-4. Hilbert's classical papers on his proof theory, stimulating as they are and transparent as they may appear at first glance, are in fact not free from ambiguity and vagueness. We do not blame him for that, of course, nor do we mean to apologize for any remaining shortcomings in our account. Nevertheless, we feel inclined to spell this out here. With the exception of Hilbert ( 1926) and (1928a) the translations of Hilbert's writings are our own. For the most part, we have slightly modified the existing translations. As to the translation of the world 'inhaltlich', we have adopted the neologism 'contentual', coined by the translator of Hilbert ( 1926), Stefan Bauer- Mengelberg. The references to the English translations of Hilbert ( 1926), (1928a) and Tarski ( 1935) appear in square brackets.