Re: Cardinality question

On Sat, 09 Apr 2005 01:24:10 GMT, "Larry Hammick"
<larryhammick@xxxxxxxxxxxxxxxx> wrote:

>"David C. Ullrich"
>>LH
>> >But I've also seen (unless I'm getting paranoid, which is quite
>> >possible) that some computer people have picked up
>> >mannerisms such as
>> >"depends on the Axiom of Choice"
>> >"in first-order logic"
>> >"constructive mathematics".
>>
>> I don't see how that indicates that there's an "anti-Cantor"
>> movement.
>Okay, "anti-Cantor" is a misleading name. I used the slogan
>"Cantor was wrong" simply because that is crank-dot-net's
>name for this category of dispute, and iconoclasts and cranks
>are always name-dropping, and in this case Cantor's name
>is the one they go for. In the same class of dispute, but
>more elementary, and not about Cantor at all, are claims
>like 0.99999... <> 1.
>>
>> >Teaching math to computers is proving to be difficult. You've
>> >probably heard of efforts at DARPA and the EU to spell out
>> >a format for mathematical results, so that theorems can be
>> >stored and linked linked just like modules in a computer
>> >program. Easier said than done, and my theory is that a
>> >resentment of things like cardinals is one of the results of
>> >frustration. But again, it might be paranoia on my part.
>>
>> Of course teaching math to computers is difficult. But
>> I don't see how that has anything to do with infinity.
>> When _people_ prove things about infinite this and that
>> they use finitely long arguments, and I don't see why
>> formalizing arguments about infinite sets so that a
>> computer can "understand" them should be any harder
>> than doing the same thing with any other branch of
>> mathematics.
>E.g. defining a specific irrational number as a certain
>partition of the infinite ordered set of rationals -- that's
>hard to "code", and there's no way any computer will
>ever verify the infinitely many inequalities that are
>implicit in that partition.

_If_ you mean that there's no way any computer will
ever verify all those inequalities by checking each
one in turn, of course that's correct, because it
would take infinitely much time. So what? There's
also no way that a _human_ will ever verify all
those inequalities, _by_ checking them one at a
time! For the same reason: it would take infinitely
much time.

Any time a human proves something about an infinite
set it's via a proof involving finitely many steps,
and there's no reason a computer cannot find the
same proof. Regarding the specific example you
mention: We might say that S is the set of all
rationals r > 0 with r^2 > 2, and T is the set
of all rationals which are either negative or
satisfy r^2 < 2. It follows that the ordered
pair (S,T) is a real number (assuming one of
various possible definitions of "real number").
Saying that (S,T) is a real number entails
that every element of T is < every element of
S. No human will ever verify that fact by
verifying s < t separately, once for every
s in S and t in T. A human can nonetheless
_prove_ that every element of S is < every
element of T. A computer can construct exactly
the same proof.

>And it gets worse, I hear, but
>you'd better ask the theoretical computing people for
>a more competent rundown than I could give.
>LH
>


************************

David C. Ullrich
.

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