Tait maintains that the chief difficulty for everyone who wishes to understand Hilbert's conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities.
At the outset of his paper, Tait attempts to pinpoint the main difficulty for everyone who wants to understand Hilbert's conception of finitist mathematics. He believes that this difficulty is embodied by the question as to how to specify the sense of the provability of general statements about the natural numbers without presupposing some infinite totality.
When he comes to describe a certain finitist framework for establishing some theses concerning finitist mathematics and PRA, he does so in terms of conditions which he himself regards as appropriate without being reasonably faithful to Hilbert.
To see whether it is sound, we must explain his conception of finitist mathematics in more precise terms.
then "finitist mathematics" in Tait's sense would be precisely explained through FIN-.
Furthermore, the conception of finitist mathematics under consideration does not provide an answer to the question as to how "[for all] x(x + 1 = 1 + x)", construed in the ordinary way, could be proved finitistically at all.
Since the interpretation of ([for all] pr) through ([alpha]) fails to square with the remainder of Tait's account, we think that what Tait has in mind when he speaks of finitist mathematics is rather captured by option ([beta]) of interpreting ([for all] pr).