Hilbert's finitism and its limits

Abstract

The use of the concept of infinite and of nonconstructive methods in mathematics needs justification. For this reason, we need to prove that these methods do not lead to inconsistencies. In order to achieve that, we formalize mathematical reasoning in axiomatic systems and prove that these systems are consistent. But in the proof of consistency, we should use only the safe methods. David Hilbert introduced the finitist standpoint to characterize this safe ground. In this thesis, I surveyed the problem of justification of non-constructive methods. Then I examined the features of finitist reasoning. I saw that the recursive mode of thought culminating in primitive recursive arithmetic is the first powerful system based on finitist principles and I surveyed its fundamental properties. Whether the finitist reasoning comprises more than primitive recursive arithmetic arises as a natural question. There are more complex forms of recursion and some seem to look as effective as primitive recursion. Following some arguments in the literature, I reached the conclusion that it is not possible to treat them as finitist.