Re: Logarithm of transfinite numbers


stephen@xxxxxxxxxx wrote:
matt271829-news@xxxxxxxxxxx wrote:
stephen@xxxxxxxxxx wrote:
matt271829-news@xxxxxxxxxxx wrote:

Randy Poe wrote:
matt271829-news@xxxxxxxxxxx wrote:

It doesn't seem right because it isn't right. The bit string
representations of natural numbers, whatever representation
you choose, consist of only a countable subset of the
uncountable bit strings of length aleph_0.
For instance
you might choose to represent the natural numbers as
...001, ...010, ...100, ...1000, i.e, infinite bit strings where only
a single bit is nonzero.

Yes, we *can* represent them this way, but we can already list them as
1,2,3,4... which we know is countable. You're just putting one bit in
the positions 1,2,3,4... so what new does this tell us?

Imagine instead that we *do* represent them in proper binary form. We
still need aleph_0 bits, and every arrangement of those aleph_0 bits
seems to map to a unique natural number.

No, every arrangement of those aleph_0 bits does not seem to map
to a unique natural number. Which natural number is mapped to
..1111111111111
or
..0101010101010


I assume ".." means a (countably) infinite series. They would both have
to be non-finite, I suppose, if that's what you mean.

Yes. It is difficult to actualy write an arrangement of aleph_0
bits in its entirety. :)

Are you saying,
very roughly speaking, that the problem is that the "number infinity"
appears infinitely many times in all those arrangements of aleph_0
bits?

I suppose that is very roughly what I am saying. More precisely,
there are lots of arrangements of aleph_0 bits that do
not correspond to a natural number. There is no "number infinity"
in the natural numbers. Every natural number can be represented
by an arrangment of aleph_0 bits that contains only a finite number
of 1's. Not every arrangement of aleph_0 bits only contains
a finite number of 1's.

So, would it be right to say that there are uncountably infinitely many
strings with aleph_0 1's, and there are no natural numbers for these
strings to map to, except, continuing my very loose way of visualising
it, "the number infinity"? In other words, we simply "run out" of new
numbers to assign to all these strings?


Only the bit strings which have an infinite number of leading
zero's correspond to a natural number, and there are only aleph_0
such bit strings.

This is where it goes wrong,
and where the flaw in the logic must lie. It seems to me that we need
to somehow show that this one-to-one mapping can't be made.

That is what Cantor did all those years ago.

Didn't he go bonkers or something? I can't say I'm surprised!

Probably because he had a vision of Usenet and the endless
threads arguing about what is in essence a very simple idea.

I disagree that it is a simple idea. I think it is a profoundly
difficult idea. In fact, I seem to remember reading that some of the
mathematicians of the day - who presumably weren't stupid - had a very
hard time accepting Cantor's arguments. And are there not still
obejectors of some standing? (I mean, not just cranks). I feel in good
company!

.

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