Re: Turing completeness of the functional paradigm?

On Fri, 15 Jul 2005, Tom wrote:
> Mr Elliot wrote:
> > > > Peano's axioms allows a well ordering, indeed a sequential ordering, of
> > > > the integers. So in this sense, Peano's axioms do impart a spacial
> > > > orientation, that the integers can be strung along a line like
> > > > cloth pins on a cloth line.
> > >
> > > I see. Still, it seems that they impart nothing else.
> > >
> > It's a natural order. Other orders my be imparted to the integers,
> > however they will not be as simple to describe within Peano's axioms.
> >
> > They could be reversed in order, they could be made into two parallel
> > lines, one of even numbers, the other of odd numbers. They could
> > be put all odd numbers first with the even numbers coming after all
> > the odd numbers.
>
> Yes, right.
>
> > > They all seem to be merely Peano sequences.
> > The Peano isn't the only instruement mathematicians play.
>
> Of course. There may be multiple names of the same rose. :-)
>
To get rational and real numbers from Peano's axioms you have to add set
theory.

> Mr Elliot, I agree. Now, please, would you be kind enough as to care to
> give me an example of a relation which is not spatial. IOW, are all
> relations spatial? I think so. But I will not assume that on my own. It
> is far far too serious.
>
The integers are infinitely countable.
Goodstein's theorem, which can be stated within Peano's axioms,
can be proved with set theory but cannot be proved with Peano's axioms.
It's a true statement about integers that can't be proved with Peano's
axioms. Thus Peano's axioms are incomplete.
.

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