$\newcommand\ZFC{\text{ZFC}}$Perhaps it would be useful to mention that set theorists have, of course,
studied numerous weaker set theories, including some extremely
weak theories, which do not give rise to higher cardinalities. One
may interpret your question as: to what extent do these weak set
theories serve as a foundation of mathematics?

To be sure, set theorists generally study these weak theories not
as foundational theories, but rather because they want to
undertake certain set-theoretic constructions in some much
stronger theory, but the objects appearing in the construction are
transitive sets satisfying these weaker theories, and so they need
to know, for example, whether those objects are themselves closed
under certain constructions. If those constructions can be
undertaken in the weak theory, then they are.

To give a few examples, the theory known as $\ZFC^-$, which is
basically $\ZFC$ without the power set axiom (but see my
recent paper, What is the theory ZFC without power set? for what this means exactly), is
widely used in set theory and has an enormous number of natural
models, including the universe $H_{\kappa^+}$, in which every set
has cardinality at most $\kappa$ and $P(\kappa)$ does not exist as
a set, but only as a class. For example, in the universe
$H_{\omega_1}$, the theory $\ZFC^-$ holds, and every set is
countable. This is a very rich universe in which to undertake
classical mathematics: you have all the reals individually, but
you cannot form them into a set; but you can still consider
(definable) functions on the reals and so on. You just cannot put
them all together into a set.

The theory known as Kripke-Platek set theory
$\text{KP}$ is another intensely studied theory, particularly for
those doing set theory with the constructible universe and
admissible set theory, and knowing what can be proved in
$\text{KP}$ and what cannot is very important in that area.

Even Zermelo set theory itself can be considered as a kind of example, since it does not prove
the existence of uncountable cardinals beyond the $\aleph_n$ for
$n<\omega$, because the rank-initial segment of the universe
$V_{\omega+\omega}$ is easily seen to be a model of Zermelo set
theory. So one could count this as a case of a weak theory that
does not prove a huge number of different infinities.

Perhaps this perspective on your question reveals that there is
really a continuum of such kind of answers. The really weak set
theories such as $\text{KP}$ and $\ZFC^-$ cannot prove even that
uncountable cardinals exist, but then slightly stronger theories,
which become true in $H_{\omega_2}$ or $H_{\omega_3}$, can prove a
few more uncountable cardinals. Zermelo's theory provides more,
but still only countably many uncountable cardinals. The $\ZFC$
theory of course then explodes with an enormous number of
different uncountable cardinals.

But let me say that this process continues strictly past $\ZFC$, for large cardinal set theorists look upon $\ZFC$
itself as a weak theory, in precisely this sense, because it
cannot prove the existence of measurable or supercompact
cardinals (and many others), for example, and so one must continue up the large cardinal
hierarchy in order to get the kinds of infinities that we like.
Set theorists consider theories all along the large cardinal
hierarchy, with the stronger theories giving us more and stronger
axioms of the higher infinite.

At every step of this entire hierarchy, starting from the very
weak theories I mentioned and continuing into the large cardinal
hierarchy, there are fundamental set-theoretic assertions that are
provable by the stronger theory but not provable by the weaker
theory.

Meanwhile, despite the fact that some every-day mathematical
objects have distinct uncountable cardinalities (and so the weak
set theories cannot prove they exist), nevertheless it is quite
surprising how close an approximation one can get just in
second-order number theory, where in a sense every object is
countable. The work of reverse mathematics generally takes place in the
context of second-order number theory, and seeks to find exactly
the theory that is necessary in order to prove each of the
classical theorems of mathematics. (Thus, they try to prove the
axioms from the theorem, rather than the other way.) They have
numerous examples of which classical theorems you can prove and
exactly what theory (provably so!) you need to do it.

a fortiori, the existence of different uncountable cardinalities) from the rest of mathematics. For instance, in analysis, establishing the Hahn-Banach theorem or the existence of the Stone-Cech compactification requires pretty much all of the axioms of set theory. To me, it's not so much that the multiplicity of uncountable cardinals comes up directly all that much in mathematics, but rather that they are inevitable byproducts of the very useful set theory axioms that we rely quite heavily in mathematics. $\endgroup$22more comments