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This may not qualify as a philosophy question exactly, but I would argue that potential applications of pure mathematics are in the bounds of philosophical interest.
Many innovations in pure mathematics, such as Riemannian geometry, turn out to have totally unanticipated applications in physics. Is this true of Cantor's work? Have his transfinite numbers had any practical applications in physics, technology, or engineering?
Not immediately. Transfinite sets have affected logic and other areas of pure mathematics, but as far as I know, they have had no practical application. Within ZF(C/U), you can fit all of the rest (ie non-transfinite) mathematics in the first two/three infinite levels.
The short answer is no, transfinite sets are not in and of themselves practical. They're certainly not "concrete" (this can be formalised by saying they're not computable).
The long answer is, there are many different approaches to the foundations of mathematics and even set theory, and ZF(C) and other transfinite theories are only a few (albeit the most dominant, for historical reasons). These transfinite theories are very powerful, and can be used as settings for the entirety of mathematical knowledge. However, there are also classical and strict finitist theories, that themselves come in various philosophical forms, and are considered sufficient to express the great majority of mathematical thought, including virtually all of 19th century arithmetic, geometry, and analysis, and much 20th century maths too. There are other foundations still that take computability as the fundamental concept, and these necessarily cannot be transfinite.
It's worth saying however that research (and even less usage) of non-ZF systems is still relatively low-key in the mathematical world. Although most working mathematicians don't even concern themselves explicitly with such low-level things as the axioms of ZF, many do occasionally resort to them, and there are set theorists who effectively study them for a living. Set theory (implicitly meaning ZF set theory) is a relatively niche area within mathematics however, and modern results from it are not generally given much attention within the wider mathematical community.
Much cryptography relies on number theory, and much number theory relies on the existence of large cardinals.
For example, the Weil conjectures (at least for curves, and presumably more generally via the same sorts of constructions) have clear immediate implications for cryptography, and the proof of those conjectures relies on the machinery of Grothendieck universes, which is to say that it relies on large cardinals.
Here are a few applications of infinite ordinals that I believe almost certainly count as practical.
Well-orders on various countable sets which are order-isomorphic to infinite ordinals are very important in computational mathematics.
The theory of Grobner Bases in computational algebraic geometry requires one to fix an ordering on the set of monomials in the polynomial ring in n variables over a field. One such ordering is the lexicographic ordering, which is of order-type n*omega, where omega is the first infinite ordinal. Real-world applications are found anywhere where solutions to systems of polynomial equations are needed.
I'd also like to argue that transfinite induction used in proofs such as Gentzen's consistency proof is in some sense finitary. It establishes that if the result in question is inconsistsent, then in finitely many steps one could produce a contradiction, since any descending sequence of ordinals is finite.
Another application that no one mentioned here is the real numbers. These form a set which is obviously infinite and in fact is uncountably infinite. People might object that in practise one never needs an actual infinity of real numbers for any particular calculation, but any finite formulation of real numbers ends up being prohibitively convoluted. That the theory of infinite sets gives us an elegant and simple framework for reasoning about the reals is a practical application in itself.
There are two theories of the infinite, the ordinals and the cardinals; neither have a direct application to the world in a concrete manner; as a generalisation of number, they have no referant as say the number two can refer to two plums.
Similarly they don't have direct practical applications in physics or engineering.
However they do have utility within the body of mathematics for example Gentzen proved the consistency of Peano Arithmetic by allowing induction to a larger ordinal.
But here practical means outside of its own domain - we're not using ordinals to solve some problem of ordinals - but an important question in some other area of mathematical logic; this sense of practical is obviously different from the sense above; and also in the sense the question uses.
I largely agree with the other answers posted, that, in a "real-science" context, there is no apparent practical applications of transfinite numbers or their arithmetic. But I'm inclined to want to come up with something, so you may consider this - and admittedly this is stretching the meaning of practical to breaking point :
In cosmology, the popular eternal inflation model makes it possible that actual infinities exist in our universe. For example, the universe could be infinite in extent, or there could be infinitely many universes in the multi-verse, or there could be a temporal infinity, etc..
If this is the case, then the question of assigning cardinals to these infinities would be one such application.
Yes and no.
Yes, in that the basic notion of cardinality is fairly important, in particular whether a set is countable or uncountable. If it's countable, you can parameterize all the elements as arbitrary-length finite strings on some finite alphabet. If it isn't, then you can't, and you need to either look at infinite strings, successively better finitistic approximations, or truncated infinite strings ending in "dot dot dot." This basic question arises commonly enough in most fields of both pure and applied mathematics that it's something you ought to know.
No, in that beyond the above fairly basic question of countability, transfinite quantities don't seem to get involved all that often. Almost all of applied math can be formalized using only three infinite cardinals - that of the natural numbers, that of the reals, and every so often, that of the power set of the reals. More to the point, most of the most powerful applied math techniques we have involve finitistic approximations of problems, such as in numeric computing, so that we are only computing things to arbitrary accuracy anyway and infinity doesn't really get involved in any meaningful sense.
The essence of the problem, unfortunately, lies in that the theory of transfinite cardinals and ordinals hasn't quite "developed" in a concrete way, similar to the rest of math, as Cantor hoped it would when he first created it. We cannot prove even basic things like if the reals are the next-largest cardinal after the naturals (the continuum hypothesis), which we now know, thanks to the results from Cohen's celebrated landmark paper on "forcing," is independent of ZFC set theory. In general, the structure of the cardinals is heavily dependent on the set theory, so to some extent you can "choose your own adventure" - yet there is no consensus on how to choose or even if there is a "right" way to choose.
As a result, it's difficult to use transfinite numbers as a core technique in applied math when much of the very basic questions are still open and buried in murky set-theoretic foundational mess.
The theory of ordinal numbers, on the other hand, does not seem to run into quite as much set-theoretic trouble right out the gate. I haven't seen them used that much in applied math (although certainly on occasion), probably because of the emphasis always being on numeric computation, which typically involves finitistic approximations of everything anyway.
There is a clear no.
Transfinite sets cannot have practical applications because of two reasons: First, there is nothing transfinite in the universe, at least in that part accessible to us. And second, transfinite set theory is a self-contradictory system, not better and not more useful than astrology.
There is a society of believers in this system who will downvote this answer and who have upvoted provably wrong answers like number theory relies on the existence of large cardinals or cryptography relies on the machinery of Grothendieck universes or Gentzen "proof" by induction to a larger ordinal or the set of real numbers which cannot be distinguished and used but have a well-order.
To prove my case for the unbiased and objective reader here is one of many blatant contradictions between mathematics applicable to real science and transfinite set theory:
Scrooge McDuck receives 10 $ every day and spends 1 $ every day, always the oldest one of his possession.
According to transfinite set theory he will go bankrupt in the infinite limit. This result is crucial for set theory. Without it, the fundamental distinction between countable and uncoutable sets would break down.
The argument is simple: For every dollar we can determine the day when it will be spent. The argument is wrong because: For every dollar we know that it is followed by at least 9 other dollars. This is overlooked intentionally because this fact prohibits the exhaustability of infinite sets - a basic requirement of set theory.
Of course in mathematical analysis there is only an improper limit: Scrooge McDuck will get infinitely rich. He will never become bankrupt, not even in any limit.
Every mathematical statement or theorm that relies on transfinite set theory is wrong or at most true by chance.
