Re: How diagonal is Cantor's diagonal sci.math 4 June 1998


In article  Nico Benschop, [email protected] writes:
>[** Erland Gadde wrote:
>The point is that this method, applied to the naturals, produces
>the string    .......111111111, with infinitely many 1:s on the left.
>This does not represent a natural number, since a natural number
>has only finitely many nonzero digits. Therefore, this doesn't imply
>that the set of naturals is uncountable.
>**]
>
>I've seen this objection before, from Dave Seaman and others, and I
>wonder where this restrictive definition comes from. In other words,
>what is the *reason* for this restriction of the naturals. I guess
>there is some consistency requirement to force this assumption/def'n.
>But it must be a *very* cogent one, to imply this a-symmetry between
>reals R[0,1) and the naturals. To an newcomer like me, it strikes me
>as a rather artificial condition on the naturals.
[]
>Mind you, I'm just experimenting in an area where I am not so
>familar with the existing theory, although I do try to get to grips
>with it (infinities) somehow. []

In article  Gilles Robert, [email protected] writes:
>This is *not* a "restriction on the naturals" : it comes from the
>definition. Things go one way and not the other
>
>- First one defines the natural numbers (Peano axioms, set theory
>  or whatever you please)
>- Then one introduces the arithmetic (addition, multiplication, etc.)
>- Then one can introduce the positional notation for numbers :
>
>The number 1234567890 for instance is *defined* as
>
>((((((((1*10+2)*10+3)*10+4)*10+5)*10+6)*10+7)*10+8)*10+9)*10
>
>If one were to allow "integers with an infinite number of digits", 
>many useful properties of the naurals would be lost, the first and 
>most important one IMHO being the induction property : in fact I
>can *prove* that every natural has a finite number of digits using
>only the induction property :
>
>- This holds for 0
>- If this holds for N, then N+1 has at most one digit more than N,
>  hence N+1 has only a finite number of digits. End of proof.
>
>If you are happy to throw out the induction property, then do so,
>but be warned that most of mathematics gets lost in the process.




Indeed this discussion does not grasp the real problem.

It seems to me that for all you (and many other people who discussed
Cantor's diagonalization previously in sci.math) it's given as obvious 
and implicit the belief that reals and numerals have necessarily some 
ontological connection.
Each one of you presupposes that reals are an ontological appendix/
derivation of naturals,
as if reals were a play of marbles 
(each marble a natural ... so we have infinite marble 
and infinite combination of marble ... dear Hesse!).

I think all this is really deeply incorrect.

History of  logic tell us that each attempt to find
an unitary foundation for  mathematics is always failed (and worst for ontological
foundation). The only thing we can accept is the Hilbert's canon: we
can/want only to be sure that our theories are consistent, complete
and decidable (when possible ... but about the principle to be decidable
there should be a lot to say ... as undecidability is again only 
diagonalization ...).

So why are we constrained to think that reals come from naturals?, 
why do we think to the reals as to be a derivation of the naturals?

Here it is the problem.

The first time when at the school we learn the definition of the reals
with respect to the naturals  the first (shocking) concept/representation 
in our mind is the line of points and the fact that points 
can *have* or *have not* a fixed distance and when
they have not is because we can think them as densely ordered.

I think the basic definition and distinction of reals from naturals could
be just here. Reals are defined by the property to be densely ordered while
naturals are not.

Reals are densely ordered, naturals are not densely ordered.

It seems clear to me that reals and naturals are defined 
by complementary property. 
If this is true, why do we persist to believe that the theory of reals 
rise ontologically from the theory of naturals?

Why we cannot think to the theories of naturals and reals as simply to two
different theories for numbers?, exactly as Euclidean geometry and Riemannian 
geometry are two different theory for the space.

Indeed I believe that the theory of naturals is the mean/structure conceived 
by the human mind for what is numerable/measurable/knowable 
(i.e. the theory when we do have "a metre", see the platinum-iridium standard 
kept in Paris, a unity of reality exhaustively defined/known, 
and hence the induction); and that the theory of reals is the 
mean/structure conceived by the human mind 
for what is not denumerable/not measurable/unknowable 
(i.e. the theory when we do not have any "metre" and hence any induction ...)

Paola Cattabriga.
[email protected]