"Credible", likely yes. Likely also awkward. To weasel out, I just said we "want completeness". When you look at the alternatives, you may conclude that you still want completeness and, thus, are stuck with the reals with no alternative you "want"!

Awkward I would agree with, but probably only because we're much more used to the usual approach (cf the "awkwardness" of Haskell).

Still, you'll see in that paper that the reals he constructs are complete. The get out clause is that they are not a set, but a class.

> The get out clause is that they are not a set, but a class.

Good grief! In axiomatic set theory, introducing 'classes' was the "get out clause" from the Russell paradox of the the set of all sets that are not members of themselves. So, the patch, the time I studied one version of axiomatic set theory, was to say that there are some sets and we know what they are and we know that the Russell paradox can't happen now, so let's go on.

It appears that the OP and alternatives to the usual treatment of the real numbers are based on what can be described. This issue doesn't impress, or, really, concern me. To me, for just an easy answer, there is the real line with points, and that's description enough for me. Apparently the OP wants a description in terms of digits and is bothered that uncountably infinitely many reals need countably infinitely many decimal digits to be described. Okay, if don't like all those digits, then just return to the points on the line and let those points be the description. Good enough for me, but at this point I have quite different fish to fry getting my business going, really like the real numbers and what classic pure and applied math do with them, and see no good reason to change the foundations of what I do.

More generally, it appears that computer science is struggling to find something to do beyond the biggies of quicksort, heap sort, merge sort, AVL trees, red-black trees, BNF, YACC, LALR parsing, relational database, P versus NP, etc. So computer science wants to take parts of statistics, optimization, control theory, and now the foundations of math for its own and, in those fields, do things differently.

Maybe I'm missing a point: Maybe the idea is to have all the 'numerical' data computing works with be describable in some finite way, that is, not just a countably infinite sequences of digits, and, thus, get rid of, say, numerical error. Maybe. Seems a bit far fetched.