Re: Relative Cardinality



Jiri Lebl wrote:

> Ahh, your wonderful "sets can only be represented by particles in the
> universe" theory. Did you ever notice that mathematicians work with
> stuff "in principle" not "in reality." Sets are IDEAS not some
> particles in the universe.

But the elements must be distinguished. How can that be accomplished?

> That is, mathematics should not change if
> we realize suddenly that all our ideas about the universe were wrong.
> I think you might be wanting to post in sci.physics, not sci.math.

No. Physicists know that already. Two elements of a set must differ by
at least one property. That is required by set theory. This property
may be such a tiny thing as the spin of one electron or a huge amount
of billions of billions of electrons and protons and neutrons building
those neurons in your brain which are required if you bother to realize
this difference as an "idea".

> Did you also ever notice that our entire theory that the universe is
> finite is based on theories that are based on mathematics that assume
> the reals are a continuum (that is complete, that is uncountable?)

No. This observation is based on finite mathematics which requires only
rational numbers, as approximations of irrational (though reasonable)
ideas.

>
> If you restrict yourself to finitistic mathematics, then you have no
> way of actually predicting that the universe is finite, so you get into
> a twilight zone where the universe could be infinite, and your argument
> against infinite sets could be wrong, but then we could use those
> infinities to predict that the universe is finite and those infinities,
> after serving us for so long would go "poof" because the universe is
> finite?

No irrational mathematics is required (see above). But your story fits
very well to set theory with its many cases which do only work if they
don't.
>
> I also thought of another irrational number that escapes your logic.
> Take the number:
> 0.9909000900000009000... That is the number whose digital
> representation has 9s in the places that are powers of 2. It is the
> "sum(n=0 to oo) 9*10^(-(2^n))". So then I can say what the 10^10^100's
> digit is, it is 0. Also the 2^100^100^100^100^100^100^100^100's digit
> is 9. And the next digit after that is 0. I wonder why this number
> doesn't exist and why it cannot be on any list.

Interesting number. Thank you for posting this question which is one of
the few non-polemic questions I received. Can you say which digit is on
the position floor(pi*10^10^100)?

I do not ask for insidious reasons or for bigotry. But it is quite
obvious that only such real numbers can exist and obey the
order-axioms, which have at least one completely well-defined n-adic
representation.

Regards, WM

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