Re: Cardinality question

On Sat, 9 Apr 2005 15:42:58 +1000, "Peter Webb"
<webbfamily-diespamdie@xxxxxxxxxxxxxxx> wrote:

>> 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.
>>
>
>I don't agree with this - I think there is a qualitative difference between
>infinite set theory and other branches of mathematics w.r.t. computers.

Of course there is, but the same difference exists wrt human
mathematicians.

Or at least that's true of all the differences I can think of.

>In most fields of mathematics, we can use the computer to model both the
>axiomatic system and models/incidences of these axioms. For example we can
>use computers to manipulate the axioms of Group Theory as well as modelling
>specific groups,

[Specific _finite_ groups - never mind, just a nit.]

>calculus as well as numerical methods, number theory as
>well as arithmetic, etc. This appears to break down for infinite set
>theory - we can use a computer to model an axiomatic system that includes
>(for example) uncomputable reals, but not the uncomputable reals themselves.

Exactly what is your definition of "model the uncomputable reals
themselves"?

Or even without giving a precise _general_ definition: Exactly what is
it that a human can do which a computer cannot, regarding the question
of modelling uncomputable reals?

>Not to mention transfinite ordinals ...
>
>


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

David C. Ullrich
.

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