Finitism

A methodological point of view, due to D. Hilbert, as to what objects and methods of argument in mathematics should be counted as absolutely reliable. The main requirements of finitism are:

1) the objects of arguments are constructive objects (cf. Constructive object), for example the written form of natural numbers, formulas in symbolic language, and finite collections of them;

2) the operations that can be applied are uniquely defined and can in principle be performed (are computable);

3) one never considers the set of all objects $x$ of any infinite collection; a general judgment $A(x)$ is a statement about an arbitrary object $x$ that one can confirm in each particular case;

4) the assertion that there exists an object $x$ with the property $A(x)$ means that one can either produce a concrete example of such an object or show a way of constructing one.

The restriction of finitism on logic is close to intuitionism, but on the whole the finitary point of view is more rigid. An argument that satisfies the requirements 1)–4) does not go beyond the bounds of intuitionistic arithmetic (see Intuitionism).

After being formalized (see Axiomatic method), substantial mathematical theories become constructive objects (collections of constructive objects). Within the bounds of the approach of Hilbert and his followers, finitism is necessary for studying such formalized theories; only those properties of theories that can be proved by finitary methods are counted as reliable. The Gödel incompleteness theorem showed that finitary methods are insufficient as foundations of mathematics. This led to the need to extend the methods that can be applied in proof theory beyond the bounds of finitism.

References

[1]	S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)
[2]	A.A. Fraenkel, Y. Bar-Hillel, "Foundations of set theory" , North-Holland (1958)
[3]	D. Hilbert, P. Bernays, "Grundlagen der Mathematik" , 1–2 , Springer (1968–1970)

References

[a1]	A.S. Troelstra, "Aspects of constructive mathematics" J. Barwise (ed.) , Handbook of mathematical logic , North-Holland (1977) pp. 973–1052