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.

,... 4, 3, 2, 1

1, 3, 5,...
2, 4, 6,...

1, 3, 5,... 2, 4, 6,...

Cantor arranged them to spread over the plane like this

1 3 6 10 15 ...
2 5 9 14 ...
4 8 13 ...
7 12 ...
11 ...
....

> They all seem to be merely Peano sequences.
>
The Peano isn't the only instruement mathematicians play.

> "The main problem of mathematics is the notion of truth." -- H.C.,
> Outlines...
>
The main problem of philosophy is truth.
This is an attribute shared by the White-Lie House.

The main problem of art is art critics.

The main problem of eternity is it takes too long.
The main problem with infinity is finding room for it.

The main problem of nothing is there's nothing to it.

Riddle of the Day: Where is everywhere?
Is it here? Is it there? Is it anywhere?
.

Relevant Pages

  • Re: Turing completeness of the functional paradigm?
    ... it seems that they impart nothing else. ... > however they will not be as simple to describe within Peano's axioms. ... > lines, one of even numbers, the other of odd numbers. ... > The main problem of art is art critics. ...
    (sci.logic)
  • Re: Turing completeness of the functional paradigm?
    ... it seems that they impart nothing else. ... >> however they will not be as simple to describe within Peano's axioms. ... >> lines, one of even numbers, the other of odd numbers. ... > Mr Elliot, I agree. ...
    (sci.logic)
  • Re: What if 1+1=3?
    ... > Wayne Throop wrote: ... > If I try to add 1+1=3 to the axioms of Peano arithmetic, ... > or 2 is odd because it is 3, I need a theory of divisibility. ...
    (rec.arts.sf.written)
  • Re: Continuum hypothesis
    ... for people to use axioms to define things when their definitions ... You cannot meaningfully have an axiom saying that 0 is odd ... one can possibly get down to ultrafinitism. ... You can't ARGUE with a DEFINITION. ...
    (sci.logic)