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

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.

 

That is true even in actual infinity.
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.

Again, totally missing the point.

You don't understand that actual infinity is a fixed quantity.

 

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.

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)]

Bijections are just as complete with "potential infinity" as with "actual
infinity".

No, that is wrong.

 

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.

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.
Regards, WM