Is there any way in which transfinite math has proved useful for the development of some technology or other?
-FrL-
“Transfinite math?” Unless you mean chaos theory…
No, he means the study of transfinite cardinals and ordinals as in modern set theory.
It’s an interesting question. I can’t think of any good response off the top of my head, though. I’d like to say that there is a lot of applications to program termination proofs, semantics of programming languages, automated proof systems, and other such things in theoretical computer science to which use transfinite ordinals can be put; however, I don’t know how much that counts as “technology”, per se (and, as that list went on, it got less and less impressive that such math might be said to be found in it).
As an example, I remember being particularly pleased one day to prove that some program of mine which used extensive continuation-based backtracking to search for solutions to certain problems happened to always eventually terminate, using transfinite induction up to omega^omega. But I doubt that is the sort of application to real, tangible technology you are looking for.
Indistinguishable is correct in suggesting automated theorem proving. Ordinals here can be used for forming termination measures on recursively defined functions. They’ve been used for program synthesis (I mention this to pre-empt any objections that ATP is simply mechanized mathematics, and not really in the spirit of the OP’s question).
Further, they’ve also found some utility in programming language semantics. See the work of Peter Hancock, for instance. This work is pretty new, so expect commercial programming languages to be using the ideas in about thirty years time 
Non-wellfounded set theory, which is similar in spirit to the theory of transfinite cardinals, is used for modeling certain phenomena. I don’t know how closely that matches up with the OP’s notion of technology, though.
It would probably be helpful for the OP to clarify both what he intends by “transfinite math” (whether he means the modern study of set theories in general, or heavy use of transfinite ordinals or cardinals in particular, or what, exactly) and what he considers to count as “technology”.
I meant the study of transfinite ordinals.
Well I’m not sure I have a clear idea of about necessary and sufficient conditions for satisfaction of the concept “technology.” But roughly, I have in mind things like this. Sometimes a technology depends on a mathematical concept in the sense that there is a machine we would never have known how to build unless we had worked out that mathematical concept. So for example, if I understand correctly, microwave ovens and computer chips both depend on the notion of an imaginary number in this sense, because each relies on quantum theory and quantum theory needs imaginary numbers.
So roughly, I was wondering if there are machines that have been built or at least imagined which we could not have understood how to build unless we had worked out how transfinite numbers work.
Having said that, people in this thread have brought up very interesting points about applications in computer programming. In a sense, writing a program is building a machine, so in that sense, this satisfies what I originally had in mind–though in a way I hadn’t quite anticipated.
-FrL-
Thinking that technology only translates to machines is kind of a narrow viewpoint. Cormen et al. make the point in the introduction to their book that algorithms really have to be considered as part of technology along with hardware and software. I haven’t seen any arguments as to whether models count as technology, but I think you could make a case for it.
I don’t think technology translates only to machines. (That’s why I put the word “sometimes” in my previous post.) For example, I think notations are technologies, I think logics are technologies, and I think games are technologies.* But in my previous post, I was trying to narrow things down in order to make them easier to think about.
-FrL-
*And I really do have only the vaguest first idea how to say what a technology is in the abstract. I’m still in the “I know it when I see it–usually” stage.
Thinking about these examples makes one wonder about a slightly different question than one focusing on technology directly. Namely, to what extent are, or could be, the concepts and study of transfinite ordinals used fruitfully in the physical sciences? And here, I can’t really think of anything off the top of my head; not that the laws of physics couldn’t have counterfactually been such that familiarity with some theory of transfinite ordinals was useful or even unavoidable in analyzing them, of course, but, to the best of my knowledge, no such situation actually has made itself apparent in contemporary physics. But I know very little about contemporary physics, so maybe someone else could provide some interesting examples of applications of the transfinite.
Yes. That.
-FrL-