David Barrett-Lennard wrote:
>
>Jesse said...
>
> > Does anyone know, are there versions of philosophy-of-mathematics that
> > would
> > allow no distinctions in infinities beyond countable and uncountable?
>I
> > know
> > intuitionism is more restrictive about infinities than traditional
> > mathematics, but it's way *too* restrictive for my tastes, I wouldn't
>want
> > to throw out the law of the excluded middle.
>
>I suggest that given any set X that is free from self contradiction, we
>can always form the power set P(X) of X (ie the set of all subsets of
>X). Cantor proved there is no onto mapping from X to P(X). Therefore
>P(X) must have a higher cardinality than X. This shows that there must
>be an infinite number of different infinities. Note however that Cantor
>used a "proof by contradiction" - so as you suggest, you would have to
>throw out the law of the excluded middle in order to allow no
>distinctions in infinities beyond countable and uncountable.
>
>- David
I guess as a mathematical Platonist my main objection is to mathematical
objects that cannot be defined in any finite way. For example, pi may have
an infinite number of digits, but we can define the notion of a Turing
machine and then come up with a finite description of a program that will
eventually output every single one of those digits. And it is possible to go
further than just the computable reals--Turing defined the notion of an
"oracle machine" which, in addition to the operations of a Turing machine,
can also decide whether any given Turing machine halts in a finite time. It
may not be possible to actually construct an oracle machine in our universe,
but the notion seems perfectly well-defined, and as a Platonist I would
think a question like "what is the nth digit of the oracle machine program
#m" must have a single true answer. And then it is possible to define the
notion of a 2nd-level-oracle machine that can decide if any 1st-level-oracle
machine program halts in a finite time, an omega-level-oracle machine that
can decide if any finite-level-oracle machine halts in a finite time, etc.
For any number that can be specified in terms of a program for a
hypothetical machine which itself has a finite well-defined description, I
am willing to believe that number "exists" in the Platonic sense.
But just based on the fact that the number of descriptions of any kind in
the English language must be countable, there cannot be more than a
countable infinity of numbers with finite descriptions of any kind.
Therefore the reals would have to include all kinds of numbers that have no
finite description at all. I am not sure I believe such things exist, and
for a similar reason I am not sure I believe that every member of the
hypothetical "power set of the integers" exists either. Am I necessarily
denying the use of proof by contradiction by doing this? Can you explain how
proof by contradiction would force one to accept the existence of
"indescribable" numbers/sets? I think the universe of mathematical objects
with a "finite description" in the general sense I describe above is a lot
larger than the universe of mathematical objects which an intuitionist would
accept, although I'm not sure about that.
Jesse Mazer
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Received on Wed Nov 19 2003 - 23:05:33 PST