If your issue is that applied math doesn't need infinite sets, then you are just plain wrong. Applied math uses infinite sets, and real numbers, and measure theory, etc. all the time. Granted, what set theorists are interested in most of the time, isn't particularly relevant for applied math. Except of course in the sense that if you one day have a question about your (applied) math which has to do with sets, it is nice that there are people who might have an answer for your question. I've had various such questions when working on Practal.