In article <firstname.lastname@example.org>, email@example.com wrote: > On Thursday, 27 February 2014 15:14:41 UTC+1, Dan Christensen wrote: > > On Thursday, February 27, 2014 6:54:55 AM UTC-5, muec...@rz.fh-augsburg.de > > wrote: > > > On Thursday, 27 February 2014 10:15:41 UTC+1, Virgil wrote: > > > > Any set satisfying the Peano postulates can serve as "the" set of > > > > natural numbers. > > > Not in mathematics. > > > This sequence satifies the Peano axioms: > > > 0, 0.9, 1.8, 2.7, ... > > > What axiom is violated? > > The axioms only specify the name of the of the first number.
> And the further numbers are not specified
> > (Even for that, we have some leeway -- usually 0 or 1). In theory, you can > > choose whatever symbols you like for the rest.
> Only if you know that they shall have the meaning known from childhood - not > from Peano.
Knowing from childhood is not at all necessary.
For example, one rarely "knows from choldhood" the sequence in base two: 1, 10, 11, 100, 101, 110, 111, 1000, ...
But it is a perfectly legitimate enumeration of members of N
> That would not be mathematics.
WM is not competent to speak for what is or is not MATHEMATICS. His expertise ceases at the boundaries of WOLKENMUEKENHEIM.
> You do not get the natural numbers unless you > specify that the differences must be units.
One may not be able to get WM's version of the natural numbers unless one plays by WM's rules, but, when playing by WM's rules one can not even get an actually infinite set of naturals like everyone else gets.
And with the von Nuemann naturals, no such a priori specification is necessary, as it follows automatically from the construction. --