  Several examples are used to illustrate how we deal cavalierly with
infinities and unphysical systems in physics. Upon examining these examples in
the context of infinities from Cantor's theory of transfinite numbers, the only
known mathematical theory of infinities, we conclude that apparent
inconsistencies in physics are a result of unfamiliar and unusual rules obeyed
by mathematical infinities. We show that a re-examination of some familiar
limiting results in physics leads to surprising and unfamiliar conclusions. It
is not the purpose of this work to resolve the problem of infinities but the
intent of this analysis is to point out that the study of real infinities in
mathematics may be the first step towards delineating and understanding the
problem of infinities in physics.
