Re: Relative Cardinality



Proginoskes wrote:

> Let me digress for a moment:
>
> Lemma. If we can't count up to N, then we can't count up to N-1.
>
> Proof: This statement is equivalent to: If we can count up to N-1, then
> we can count up to N. This is trivially true.

No. Use, as a simple model, three chips. How far can you count and
store the result by means of these chips? 1 , 2, end. In order to store
3, you must forget 1 and 2. Now turn the whole universe into a big
computer. The principle remains the same, the numbers get larger
though.

> > > Er ... Euclid proved that there are an _infinite_ number of primes.
> >
> > Euclid did not talk of infinity. There are more than any given number,
> > he said.
>
> That's what infinity means.

No. By this definition the values of the natural numbers would become
infinite. But they don't by definition of natural number.

> Actually, he didn't say, "There are more
> than any given number," because he didn't know English.

A Greek word to use was available: apeiron. He did not use it
deliberately.
>
> > > What do you find wrong with that proof?
> >
> > Do you really think it necessary to demonstrate such things here?
>
> That's not what I asked. I asked what YOU found wrong with that proof.

The proof is correct, in principle. But it is impossible to form
numbers which require more than 10^100 different digits which cannot be
derived from a simple rule.

> If that's what you've really meant, then you should have said so,
> instead of wondering about whether numbers of the form 111...111 give
> you infinitely many primes, since you won't be able to write them down
> anyway.

Not write them down, but I would know each digit of that number.
Therefore, for any other number I could find out which of them is
larger. This is the criterion of existence for a "measure of
largeness", i.e., a number.
>
> Mathematics was designed to work outside of the physical universe.

Nothing does work outside of the physical universe.
>
> > Therefore
> > there cannot exist a set with more than 10^100 elements and we cannot
> > count up to 10^10^100.
>
> Well, this will make life much more difficult for you as a physicist,
> since physics uses lots of statistical mathematics (which allows sets
> of arbitrarily large size, even sets of size 10^10^10^10^10^10^100) to
> make certain calculations easier.

Those numbers do certainly exist. But how would you satisfy the
set-theoretic requirement that each element must be distinguished by at
least one property from its companions? With less than 10^100 particles
in the whole universe?

Regards, WM

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