Let's say you are a prospective mathematician with some addled ideas about cardinality.
If you assumed that the natural numbers were finite, you'd quickly vanish in a puff of logic. 🙂
If you thought that natural numbers and reals had the same cardinality – measure theory would almost surely break down, and your assumption would conflict with any number of "completeness theorems" in analysis (like the Baire Category Theorem for instance).
However, let's say you concluded that there were only three types of cardinality – finite, countably infinite, and uncountable.
Would this erroneous belief conflict with any major theorems in analysis, algebra or geometry ? Would any fields of math – outside set theory – be clearly incompatible with your assumption ?
PS: Apologies for the provocative title. Hope the question is clear.
Best Answer
$\newcommand\ZFC{\text{ZFC}}$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.