In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 29 Mai, 06:44, "Peter Webb" <webbfam...@DIESPAMDIEoptusnet.com.au> > wrote: > > "WM" <mueck...@rz.fh-augsburg.de> wrote in message > > > > news:email@example.com... > > 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?
WM's suggested system is provably inconsistent so his suggestion is about as useful as tits on a bull, whereas ZF has no known inconsistencies that are logically derivable without imposing other assumptions.