Your "well-defined" is confused. Enumeration and identification are two different things, we can identify a human when presented with one, but not enumerate all humans by a formula. "Concrete definitions to support formal existence" are redundant because once we have an identifying characterization (e.g. by axioms) we can draw conclusions about any object that falls under it without any handwaving. There is no need to ever present even one such object, let alone all of them. And if "not well-defined" equaled "absurd" there would hardly be a non-absurd word in all of English, I fear.

@Conifold Why is it not the case that we are taking an 'undue' advantage of "appeal to intuition" of Real numbers to identify them?

Honestly, the real numbers are some pretty weird objects, and I'm not convinced that "real" is the right adjective. The vast majority of real numbers cannot be described or defined using any combination of notations from published mathematical journals, textbooks, etc., because such notations can be Gödel numbered and therefore enumerated. Similarly, for the vast majority of real numbers, you can't write a proof that the number is not equal to zero, because proofs can also be enumerated. But they are topologically and analytically convenient, so you can't just discard them altogether.

@Kevin So what? Black holes are pretty weird objects and the vast majority of them is forever beyond our observational horizon, that does not stop us from having a theory of them. @ Ajax Appeal to intuition is only needed to formulate the axioms, which is, by nature, a duly handwavy activity. Once you have the axioms intuition can be dispensed with, along with actually identifying anything they describe, for that matter, as far as mathematics is concerned. Only interpreting it for applications needs actual identifications, but not every mathematical entity need to be matched to a real one.

@Conifold: Well, personally, I'm skeptical of any numbers being "real" in the usual (physical) sense. It's a very ambiguous adjective, and mathematics is supposed to favor precision. See also physicists using the term "realism" to refer to hidden variable theory, which is problematic for much the same reason.

@Kevin Mathematics does not favor "precision" in labels, it ignores them. They are just as much historical accidents as most labels everywhere else, and irrelevant as far as the content of theorems is concerned. "Real" is no more particularly connected to realism than "rational" is particularly connected to reason. We got "rationals" because Pythagoreans had their numerology, and we got "reals" because Descartes thought they were more real than imaginary roots. But the use in mathematics is perfectly unambiguous.

@Conifold: Yes, I am in fact arguing semantics. I had thought that was obvious.

I wonder why this question has been downvoted. After all @Kevin summarized the case that the reals are absurd. There's nothing real about them, and even their mathematical existence is strange. For example there are countable models of set theory (if there are any models at all) in which the reals are uncountable merely by virtue of there not being any bijections between them and the naturals, and not for any intrinsic reason. The real numbers are most definitely absurd, which is not to say they're not useful. So why the downvote?

@user4894 The question has been voted/voted closed largely because of the idiosynchratic syntax which creates a lack of clarity. As Conifold noted, the choice of diction confuses several concepts that are well-recognized by the formally trained.

Edited to avoid the "clarity objection".

There are two ways to respond to such questions: 1) by rejecting questions based on well-formedness and 2) by responding to the substance of the lack of well-formedness. Many contributors prefer the first method. There are several excellent responses below that provide formalisms and clear up the misconceptions of the OP.

@Kevin Whether or not you believe and use "real" to describe numbers has little impact on it's established convention. You can either use conventional language and be understood or idiosyncratic language and be misunderstood. See the difference between presciptivism and descriptivism.

@JD If lack of clarity counts against philosophers, what do you make of Hegel? Do you agree that there's a case to be made that the reals are absurd? By the way I don't see that the question's been closed. And I hope it's not. People should understand that in fact the mathematical real numbers are quite absurd. Useful as all get out, to be sure. But absurd. It's a perfectly good question.

@user4894 I struggle to find any case for "absurd" either in Kevin's or your comments, or even what "absurd" means. Would they be less "absurd" if we called them "usefuls" instead of "reals"? There are models of Peano arithmetic with infinite natural numbers, does it mean that natural numbers are "absurd" too? This is a general phenomenon with formalizations in first order logic, they do not fix the intended model, it has nothing to do with real numbers in particular. If there is a case for "absurd" it would help to spell out what that means and base it on something specific to reals.

@Conifold Thanks. I would replace "absurd" with "much less concrete than nonspecialists generally believe," would that be fair? Your remarks precisely capture my intent, the non-categoricity of first-order models; along with intuitionist doubts about R. "Absurd" is too strong; but I do want to counter the viewpoint expressed by @J D that these matters were "resolved in the 1870's." I'm reading through Putnam's article on models and re-reading Feferman's paper on CH, and of course despairing of writing anything that would meet your standards. But I think you, at least, perfectly get my point.

@Conifold In terms of arguments specific to the reals, kevin outlined those issues, which also go to the "much less concrete than nonspecialists think" argument.

@user4894 I see that you don't like the real numbers. Wait till you hear about ℂ, hyperReals, and the arm−chair postulates. xD

@user4894 I thought you meant some loose sense of "paradoxical", as in highly counterintuitive, but was surprised that the usual suspects were not brought up. As for concreteness, every individual real number is exactly as concrete as the set of natural numbers, which people seem to be very comfortable with (perhaps unjustifiably). It is only some sets of real numbers that can be blamed for non-concreteness. But then we do not expect much conreteness from sets of anything, they are more abstractions circumscribed by predicates. And reals are not wedded to FOL, SOL version is categorical.

@BertrandWittgenstein'sGhost I love the real numbers and am familiar with the complex numbers and the hyperreals. The arm-chair postulates are new to me, what are those? It's like loving chess, but realizing it's only a formal game and has no real-world instantiation. I don't reify the real numbers.

@user4894 I am messing with ya. You hit the nail on the head. I would call all these number games just that: arm−chair postulates. Useful? Sure. Meaningful... Well that's for the realists to decide.

@Conifold Not concerned with the usual paradoxes. It's not that the reals are counterintuitive. It's that they're too intuitive. Everyone believes in them as a model of continuous time, say. But physics doesn't support such a view. The reals are not (as far as we know) instantiated in the physical world. There aren't any infinite sequences. Those are the points kevin brought up in a comment. I don't agree that "every individual real number is exactly as concrete as the set of natural numbers," Every real number is equivalent to some set of natural numbers, but if the set is noncomputable ...

@Conifold (... cont) then the constructivists and intuitionists object to its existence and perhaps they have a point. "we do not expect much conreteness from sets" -- I am mostly reacting to J D's 1870 remark. Sets are not nearly as coherent as people think, mostly for Lowenheim-Skolem reasons. Feferman's paper "Is the Continuum Hypothesis a definite mathematical problem?" makes this point, as does the Putnam paper I linked earlier. My understanding is that logicians don't like SOL but I don't understand this issue very well.

@user4894 4th vote for closure is in, so I suspect this post is doomed. Let me briefly respond. I see the realist-instrumentalist dichotomy as well as the realist-constructivist dichotomy as false dilemmas each because I approach these issues from a process philosophy perspective that inherent in my metaphysical positions which I shall disregard. I just read Putnam's paper, and I believe his "Skolemization of everything" is absolutely a philosophy of language issue, and is reflected in the same issues surrounding Duhem and Quine and underdetermination and holism. As Conifold says...

reals are as naturals, though I disagree that it is unjustifiably "real" because reality is often used to describe direct realism, and not more broadly. I believe there is much to be gained out of transcendental idealism (Kant) and embodied realism (Lakoff and Johnson) in dissolving the debate, although I won't go into particulars...

I reject some traditional metaphysical notions surrounding models and sets as things rather than processes, and consequently find strengths in both seeing sets as things, and seeing sets as constructions, and believe that intuitive notions play a role in accepting that meaning doesn't inhere to the axiomatic schemes and rules of inference, but lie beneath them in metaphysical presumptions of language and experience, or if you prefer Wittgensteinan language games.

@Conifold And I think you're right to attack "absurd" as being more connotation than denotation and certainly not theory. At best, it's exclamatory and is meant to relieve the anxiety from the cognitive dissonance that is triggered from the way the Continuum forces our mind to cope with infinite regress.

@user4894 It was unfair to say that the matter was settled broadly, because extreme Platonic thinking and extreme Constructivist thinking are popular; but said popularity to me is part of the, let's say, Meinongian jungle of the metaphysical presupposition that extends right into natural theology and divine revelation. You're right to have tilted me from my high horse, sir/madam!

@Conifold And of course, sir/madam, I thank you for your commentary and recommendations. I found Etchemendy's work on the logical consequence in line with my personal preferences to see the limitations of the formalisms of syntax and deduction.

Gentleman/women, thank you for most excellent commentary.