Eckard, I'm glad you enjoyed reading the thread on Dedekind's axiom: the style of Canute is excellent. I think many here in FQXi forums and contests could learn something from him (including myself, of course).
I just downloaded Prof. Mückenheim's sourcebook. It seems very rich and interesting, certainly very useful for me, regardless of whether or not I agree with the conclusions.
You say: "I see Dedekind and G. Cantor having established the impossible: a continuous line that is composed of distinguishable from each other points."
It seems to me that your words express, referring to the linear order of the real numbers, the meaning of the axiom of choice. Gödel and Cohen demonstrated its independence from the other axioms of Zermelo and Fraenkel. Many mathematicians accept it (more or less implicitly). Others prefer to do without it (usually explicitly). It may be that it is an expression of our defective or erroneous way to consider the real numbers. But it may be that it reflects some deep aspect of their reality and of the nature of mathematical infinity, an infinity that we are not able to completely grasp.
I agree with you that perhaps we will never know if space and time are finite or infinite. In fact we don't know, after 2500 years of theory and research, what they are. And we don't even know what the numbers are, if they are discovered or invented, if they exist outside the mind or are just a product of it, if they are dicrete or continuous, if they are in a sort of Plato's hyperuranium or of Cantor's "paradise" (or Cantor seen by Hilbert's eyes).
I don't think that all sets of numbers we know and use are mental constructs. So are many of them, like the infinite hierarchy of Cantor's transfinite, and probably also imaginary numbers. I tend to think, however, that there is a link between the natural numbers, the positive real numbers, and the world. I find it hard to think, for example, that Pi is only the result of mind's creativity.
In the 2015 contest, I have proposed the hypothesis that real numbers (suitably ordered), space, and time are the same thing, at least for all the reference frames that travel at speeds below that of light. I am not able to prove this hypothesis, nor to deal on my own with all the complexity of the issues it raises. But I think I can argue (as I did in a book and partially in the current contest), it allows to capture som aspects of the nature of time (as long as we exclude the axiom of choice) and probably to explain the possibility of motion and change, very common phenomena that have been always a source of difficulties and paradoxes.
I think I have dwelt too much in this post. But I think also it is a pleasure to converse with you, Eckard.
Best regards again,
Giovanni