On 29 Mai, 06:44, "Peter Webb" <webbfam...@DIESPAMDIEoptusnet.com.au> wrote: > "WM" <mueck...@rz.fh-augsburg.de> wrote in message > > news:firstname.lastname@example.org... > 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. > > ************************* > > How do you falsify the existence of the set of all Natural numbers in ZF ? > ZF includes an axiom of infinity, which pretty much directly guarantees that > there is an infinite set of all finite ordinals.
What about classical arithmetics with an axiom that sqrt(2) is a rational number?