  The stipulation that no measurable quantity could have an infinite value is
indispensable in physics. At the same time, in mathematics, the possibility of
considering an infinite procedure as a whole is usually taken for granted.
However, not only does such possibility run counter to computational
feasibleness, but it also leads to the most serious problem in modern physics,
to wit, the emergence of infinities in calculated physical quantities.
Particularly, having agreed on the axiom of infinity for set theory -- the
backbone of the theoretical foundations of calculus integrated in every branch
of physics -- one could no longer rule out the existence of a classical field
theory which is not quantizable, let alone renormalizable. By contrast, the
present paper shows that negating the axiom of infinity results in physics
acting in a finite geometry where it is ensured that all classical field
theories are quantizable.
