On 27 Mai, 23:15, Virgil <virg...@nowhere.com> wrote:
> > > > Of course there can be different models for N and there can be > > > > different names for the elements of N. But that does not matter. The > > > > natural numbers do not enter mathematics because someone "defines" > > > > them, names them, or makes models of them, but because the natural > > > > numbers are simply existing and mathematics is built upon them. > > > > But according to WM, no such thing as N can exist. > > > So WM wold throw out the naturals on which so much is built. > > > The question is not whether the complete set of all naturals exists. > > That is a question on which WM differs from the mainstream. > But WM cannot grant its existence in a given argument and then deny it > in the same argument and expect anyone to accept that sort of argument.
Grant the existence of two natural numbers m and n such that m/n = sqrt (2). Then falsify it. Grant the existence of a largest natural. Then falsify it. Grant the existence of a largest prime number. Then falsify it. Grant the existence of all natural numbers. Then falsify it.
All these proofs are proofs by contradiction. > > > That question alrady is nearly as ridiculous as any affirmative > > answer. > > Wrong!
I said "nearly"! This is a fuzzy quantification, not a sharp one. It cannot be wrong. One might qualify it as nearly wrong or nearly right, at most. > > > > > The question is whether we could inform someone who does not yet know, > > what we understand by the sequence of natural numbers. > > Children know from a very early age. Whom does WM think is left to > inform?
How do children get to know what the real numbers are? Do they have Peano's axioms in their genes or in their strollers? > > But non-static sets do not exist in any mathematical set theory.
That is orthodox nonsense. The natural numbers as a potetntially infinite set exist in every mathematical theory before BC (before Cantor) and in many mathematical theories AC, for instance in every theory that is taught in elementary schools as well as in universities outside the departments of "logics" and mathematics.