Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Infinity: The Story So Far
Replies: 16   Last Post: Mar 7, 2014 8:31 AM

 Messages: [ Previous | Next ]
 Virgil Posts: 8,833 Registered: 1/6/11
Re: Infinity: The Story So Far
Posted: Mar 4, 2014 5:01 PM

mueckenh@rz.fh-augsburg.de wrote:

> On Tuesday, 4 March 2014 21:47:52 UTC+1, Virgil wrote:
> >
> > mueckenh@rz.fh-augsburg.de wrote:
> >
> >
> >

> > > But he claimes that the Peano-axioms supply the natural numbers of formal
> > > mathematics. So the natural numbers of formal mathematics are:
> > > 1, -10, 100, -1000, ...

> >
> > Elsewhere, the successor operation is denoted by adding one.

>
> Could you please quote the axiom?

Did I say that it was done in an axiom?
But it is quite standard in interpreting the Peano axioms.
http://mathworld.wolfram.com/PeanosAxioms.html
Peano's Axioms
1. Zero is a (natural) number.
2. If n is a (natural) number, the successor of n is a (natural) number.
3. zero is not the successor of a (natural) number.
4. Two (natural) numbers of which the successors are equal are
themselves equal.
5. (induction axiom.) If a set of (natural) numbers contains zero and
also the successor of every (natural) number in it, then every number is
in it.

Note that successorship is nowhere defined to be adding one within the
axioms themselves. It is a definition made separately.

SO that WM is wrong again!
--

Date Subject Author
3/4/14 Virgil
3/4/14 Virgil
3/5/14 Virgil
3/6/14 Virgil
3/5/14 Peter Percival
3/5/14 Virgil
3/6/14 mueckenh@rz.fh-augsburg.de
3/6/14 Virgil
3/7/14 Peter Percival
3/6/14 Peter Percival
3/6/14 Virgil
3/7/14 Peter Percival