Am Montag, 6. November 2017 08:21:29 UTC+1 schrieb bassam king karzeddin: > Why do we really need those real non-constructible numbers, if it is impossible to express them exactly except only by constructible numbers or as meaningless notation in mind only?
Neither does anyone need them nor can anyone use them. At least not in mathematics. The only occupation for such non-existing creatures is matheology.
"how can one assert that something like the continuum exists when there is no way one could even in principle search it, or even worse, search the set of all subsets, to see if there was a set of intermediate cardinality? Faced with these two choices, I choose the first. The only reality we truly comprehend is that of our own experience. But we have a wonderful ability to extrapolate. The laws of the infinite are extrapolations of our experience with the finite." [Paul J. Cohen: "The discovery of forcing", Rocky Mountain Journal of Mathematics 32,4 (2002) 1099f]
Most of these extrapolations, starting with the use of the bijection as a measure for infinite sets, are absolutely unscientific because (1) really scientific results will never depend on "clever choice" of indices, and (2) universal quantification over infinite sets shows that every element is followed by infinitely many elements, proving that universal quantification is impossile per se.