One of the most often heard arguments in favour of transfinite set theory is the completeness requirement of |R and real functions. It is not true.
Well, then tell me, Herr Professor Doktor Mueckenheim, how do you solve the equation ih partial(du/dt) = H(u) [JR, Matheology § 022, sci.logic June 13, 2012]
In general, yes, most real numbers lack names, and we cannot effectively distinguish them (in the usual story). [AS, Matheology § 022, sci.logic June 13, 2012]
Numbers are free creations of human mind. They serve as a means to easen and to sharpen the perception of the differences of things. Zahlen sind freie Schöpfungen des menschlichen Geistes, sie dienen als ein Mittel, um die Verschiedenheit der Dinge leichter und schärfer aufzufassen." [Richard Dedekind: "Was sind und was sollen die Zahlen?" 1887, 8. Aufl. Vieweg, Braunschweig 1960, p. III]
But if not even the numbers can be distinguished, what are they good for? Real numbers that cannot be distinguished will not complete mathematics, they will not make the real axis continuous, they cannot guarantee that every polynomial has its zeros. All real numbers that ever can appear in mathematical calculations have finite names (at least the definition of the problem like: "find the fourth root of 16") and belong to a countable set. Therefore uncountably many unreal reals are good for nothing.
I am convinced that the platonism which underlies Cantorian set theory is utterly unsatisfactory as a philosophy of our subject [...] platonism is the medieval metaphysics of mathematics; surely we can do better. [S. Feferman: "Infinity in Mathematics: Is Cantor Necessary?"]
Feferman shows in his article, "Why a little bit goes a long way. Logical foundations of scientifically applicable mathematics" on the basis of a number of case studies that the mathematics currently required for scientific applications can all be carried out in an axiomatic system whose basic justification does not require the actual infinite. http://www.hs-augsburg.de/~mueckenh/GU/GU11.PPT#416,62,Folie 62
Though Gödel has been identified as the leading defender of set- theoretical platonism, surprisingly even he at one point regarded it as unacceptable. In his concluding chapters, Feferman uses tools from the special part of logic called proof theory to explain how the vast part if not all of scientifically applicable mathematics can be justified on the basis of purely arithmetical principles. At least to that extent, the question raised in two of the essays of the volume, "Is Cantor Necessary?," is answered with a resounding "no." [S. Feferman, loc. cit, Description from the jacket flap] http://math.stanford.edu/~feferman/book98.html