Re: Relative Cardinality


mueckenh@xxxxxxxxxxxxxxxxx wrote:
> 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.

That's right, you're not allowing any numbers with more than 10^100
digits, because you can't write them down in the real world. How
pathetic; you haven't even scratched the surface in what numbers are
allowed outside this universe.

(Comment added later: The hydrogen atom has an infinite (countable)
number of stable energy states. Any amount of energy can be created, as
long as it's "paid back" a short time later, so you can store one of a
countable number of natural numbers in one atom.)

> > > > 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.

Not this @#$%@ again.

Every natural number is finite, since it only has a finite number of
digits. But there are an infinite _number_ of them.

That's like saying that {4, 6, 8} is a set of prime numbers, because
the size of that set (3) _is_ a prime number.

> But they don't by definition of natural number.

Actually, they don't by _your_ definition of a natural number (it has
to be less than 10^10^100). The definition of natural number allows
more digits than yours does.

> > 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.

You missed the point of what I said. You were arguing over the exact
words he said, but you weren't even close because the words were in
English.

I figured everyone would know there was a 8-) at the end of that
sentence.

> > > > 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.

What do you mean by 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.

You need to record the number of digits in the number 111...111,
though, because 11 and 111 differ only in the number of digits. That
means you can't deal with numbers with that "simple rule" which are
larger than 10^10^10^100.

But this means that, if you came upon the number N (> 1) written among
all the particles in the universe, how do you decide whether it's meant
to be N or 111...111 (with N digits)?

This means that your system of recording numbers on particles is not
"well-defined", because you can get more than one number out of a
single representation.

> 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.

How can you be sure of this? All the physics that has been developed
has been for use in this universe, based on observations and trying to
come up with a system that fits those observations.

There's no problem with my statement, because I included the phrase
"was designed to."

> > > 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.

So you've stated that there are numbers that exist that you can't write
down.

But this has been your point throughout the discussion. So you've
contradicted yourself here.

> 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?

I don't need to follow the second requirement with "my" mathematics,
which is the point of the whole thread. _You_ need to tell me this,
because you've insisted on it.

--- Christopher Heckman

.