In article <ee23e72e-206f-48c9-ac34-ad7b2046f299@c9g2000yqm.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> 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.
I have never seen a logically correct or complete falsification of the existence of all naturals.
Note that an argument against the existence of numerals (names) for all naturals does not suffice. > > > > > 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.
All of WM's logic is fuzzy, much to fuzzy for serious mathematics. > > > > > > > > > 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?
Quite possible in their genes. > > > > But non-static sets do not exist in any mathematical set theory. > > That is orthodox nonsense.
Then produce an example of a mathematical set theory is which there are non-static sets.
> 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.
Balls! Set theories as taught in elementary schools almost all require a fixed universal set in which all other sets are fixed proper subsets, a la Venn diagrams. And this approach even carries over into some introductory math classes in universities.
Certainly I have never seen such variable sets anywhere in either England or the United states at any level from kindergarten through postgraduate. > > Regards, WM
