Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

Am Mon, 07 Oct 2024 09:41:23 +0200 schrieb WM:

On 06.10.2024 17:48, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

 

This idea of time may be what misleads the mathematically less adept
into believing that 0.999... < 1.

We can add 9 to 0.999...999 to obtain 9.999...999. But multiplying
0.999...999 by 10 or, what is the same, shifting the digits 9 by one
step to the left-hand side, does not increase their number but leaves
it constant: 9.99...9990.

Totally irrelevant to my point.  I was talking about the unbounded
sequence 0.999....  You have replied about a bounded finite sequence of
9's.

No, even an unbounded sequence does not get longer when shifted by one
step.

Nor does it get shorter, it stays infinite.

Set theory depends on actual infinity.

How would it go wrong if there were merely potential infinity?

Bijection means completeness. Potential infinity is never complete. But
potential infinity is used in fact, best seen with Hilbert's hotel or
mapping of natural numbers on even natural numbers. It is the reason why
all countable sets are countable. I actual infinity there are more
natural numbers than even natural numbers.

Bijection is not about completeness, countability is. Of course stopping
after a finite number, which potential infinity seems to mean, is not
„complete” in that sense. Hilbert’s Hotel is actually infinite, it
already holds infinite guests. All of them can at once move to the next
room, no need to do it stepwise (couldn’t you start with the „last”?).

Bijections must be complete.

The word "complete" is misleading when talking about infinite things.

No. Completeness is required. Cantor claims it: "The infinite sequence
thus defined has the peculiar property to contain the positive rational
numbers completely, and each of them only once at a determined place."
[G. Cantor, letter to R. Lipschitz (19 Nov 1883)]

Then you are talking about actual infinity.

But Cantor's bijections never are complete. Cantor's list must be
completely enumerated by natural numbers. The diagonal number must be
complete such that no digit is missing in order to be distinct from
every listed real number.  Impossible. All that is nonsense.

Yes I agree with that last sentiment.  Talking about "completely" with
regard to infinite sets is nonsense.

Then bijections are impossible.

*between infinite sets, for you

It isn't even clear what you mean by saying the diagonal number must be
"complete".  It's generated by an infinite process, but remember
there's no time involved.  It just is.

If it is never complete, then always more is following and the diagonal
number is never excluded from the list.

It is completely infinite.

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.