Re: Cardinality question


> 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.
>
>
> Exactly what is
> it that a human can do which a computer cannot, regarding the question
> of modelling uncomputable reals?
>
> David C. Ullrich

You are right, of course. If we are all just Turing machines, then there is
no difference. And the margins of Usenet are not big enough for me to prove
that we are NOT just TMs.

However, my point seems fair enough in context of the original post -
essentially, what use is there for theories of infinite sets, beyond a few
impossibility proofs in Computer Science that 95% of programmers would never
have heard of ...

The OPs argument is that infinite set theory is useless. This appears true
to me, because any describably physical system can only have a countable
number of states.

However, I am also aware that through the 18th and 19th centuries number
theory was regarded as the "Queen of Mathematics" because it did not have
(and people believed it could not have) any practical use. This was a
failure of their imaginations - shame on Fermat for not considering internet
cryptography when he was investigating primes. Perhaps infinite set theory
does have some purpose, and there is some phyiscally realisable model that
the theory relates to, and this is just the failure of our imaginations when
we consider it somehow different from the rest of mathematics?






.

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