Subject:      Re: Allright, how many reals *are* there?
Author:       Nico Benschop 
Date:         6 Nov 98 04:39:17 -0500 (EST)
Newsgrp:      sci.math
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Jeremy Boden (mailto:jeremy@jboden.demon.co.uk) anwered Graham Fyffe:

Remember when you are performing Cantor's proof that you are free to
extend a number with as many zeroes as you want. You don't have a
problem with numbers only being 'd' digits long. This does allow you
to build a "square" array of digits, which may help with the
visualisation of the diagonalisation.

BUT...
  What does 10^aleph_0 mean?
  How do you know it's uncountable?
  Is it the same as 2^aleph_0?
  What does that actually mean?
  Hint: Think of power sets.

Cantor's theorem doesn't actually say anything about constructs like
10^aleph_0; it simply gives a construction for which *any* countable
list of reals can be shown to be incomplete. Hence |R| > aleph_0. (#)

Alternatively, R is not countable, or: R is uncountable.    ....(*)
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NB:   Re:  (#) vs (*)
Now this is interesting: Cantor's approach is a matter of CHOICE
about the "reals" (re: infinitely divisible intervals), isn't it!?

All those counter intuitive conclusions about uncountable infinity
follow from a CHOSEN interpretation -- based on an argument stemming
 from a (decimal or other base) REPRESENTATION that from the start
is assumed to be inadequate;-( ... since, as Graham (and I and many
others) have pointed out that Cantor's "diagonal" CANNOT possibly be
"pulled up" from finite representations -- via induction on "d",
the number of digits which is necessarily countable. So WHY start with
such representation in the first place!?


 Re: How diagonal is Cantor's diagonal?  ...(sci.math 6jun98) 

It is like wanting the impossible from a blind man:

  "Look, here is a beautiful landscape: even if you can't see it
   just _imagine_ it .... proving that it does not exist, since
   you'll never be able to disprove me: yet you
   can play imaginary games in it, can't you?"

If he likes such games, fine: he'll follow the suggestion (to give him
something to do - very useful _also_).
Could it be that Cantor wanted some infinity beyond the Naturals so
desparately that he came up with this "bootstrap" trick? (nota bene
based on representation;-)

Fine with me, but who will swallow it? Cerainly not those with a
view of Math as a tool to understand the world around us, AND in us.

By now, in the digital computer area, people are used to, and insist,
upon representing the objects (here "reals") which one is supposed to
operate with. Such "gedanken experiments" as Cantor's have no appeal
anymore, IMHO. Moreover, there is no NEED for uncountable infinity...
    Really, WHAT practical model of our world requires it?
The countable infinity of the Naturals is nec&suff for induction,
which is nec&suff for all constructable concepts & objects [including
limits, as required for concepts like pi, e, sqrt(2), Liouville's
trancendental number T = \sum 2^{-n!} (for all integers n>0)... etc,
etc.]

Concepts such as "density" and "infinitisimal" do not really require,
(for all practical purposes;-) some uncountable infinity(ies), do
they? And to realize that 2^k > k the uncountable reals are bit
overdone, to put it mildly...

Somehow, talking of 'orthogonality' in this NG, these endless
discussions seem to be more of a psychological nature, of the
"game-players" vs the "constructors": it is a matter of choice/taste
where you put your priority (at the bottom, building upwards; or at
the top, floating even further away) _and_ what you accept as being
"proof".

I agree that other "imaginary" math concepts, like negative numbers,
and i (with i^2 = -1), and Fourier's frequency spectrum, had initial
trouble to be accepted. But eventually they _were_ generally accepted,
because they turned out to model nicely and efficiently the world
around us -- Notice that they were born from a practical NEED,
which IMHO cannot be said of Cantor's gedanken experiments. Unless of
course you admit playing with uncountable infinities to be a natural
need... Again, it remains a matter of taste (and in my opinion it will
remain so "forever" -- at least I have a hard time to imagine some
_practical_ need & application of uncountable infinities) -- As has
been shown over & over again: matters of taste are ideal for endless
discussions...

Ciao, Nico Benschop 
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