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From: David Bernier <david...@videotron.ca>
Newsgroups: aus.tv,sci.math,sci.logic
Subject: Re: New Anti-Cantor Attack
Date: Sat, 05 Jun 2010 02:32:08 -0400
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Aatu Koskensilta wrote:
> David Bernier<david...@videotron.ca>  writes:
>
>> Feferman also discusses ACA_0, a theory having to do with
>> arithmetic-sentence based analysis or something like that.
>
> ACA_0 is a conservative extension of PA in the language of second-order
> arithmetic -- that is, all of its theorems which can be expressed in the
> language of PA are provable in PA, and conversely. It has both number
> and set variables, and as axioms induction stated as a single sentence:
>
>   For all sets X, if 0 is in X, and n+1 is in X whenever n is, then all
>   numbers are in X.
>
> the usual axioms for successor, addition and multiplication, and a
> predicative comprehension schema:
>
>   For any formula P containing no bound set variables, the universal
>   closure of
>
>    There exists a set X such that a number x is in X if and only if P(x).
>
>   is an axiom.
>
> This theory is predicatively justified, in the sense that we can make
> predicativist sense of its axioms, and can provide arguments for them
> that are compelling on the predicativist conception of mathematics. (It
> is also finitely axiomatizable, unlike PA itself.)
>

Thanks for your explanation.  Poincare wasn't a slouch, and neither
were Weyl or Brouwer.  So I try to grasp their point of view,
to the extent possible.

Admittedly, there are many existence proofs today which use the
Axiom of Choice, and reals can't all be listed.

The other side of the coin for me goes like this:
I like to think (picturing the Cantor set), that any ternary
expression  0.b_1 b_2 b_3 ... b_j ...   1<=j< oo ,
b_j in {0, 2} makes sense.  Then there is measure theory according
to which every countable subset X of the reals has Lebesgue measure zero.
(And so on).

All of Lebesgue measure theory in R^n, compactness and connectedness
(general topology), Baire category, make sense to me.

So I'm sort of disappointed with what I read of Poincare in that
for me, it doesn't seem to give a fully satisfactory
account or description of the continuum (the reals, the real number line):
what is there to find in the real number line? What reals exist?

And, BTW, if you happen to know what Weyl and/or Poincare thought
of Dedekind cuts and/or Cauchy's development of Cauchy sequences,
what with "every Cauchy sequence converges", I'd be happy to
hear about it.

David Bernier