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

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

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.

The concept of "length" appropriate for finite sets doesn't apply to
infinite sets.  Think!  infinite means "without end" - unendlich.  You
are manipulating the supposed end of an endless sequence.  Nonsense!

Again, totally missing the point.

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

It may be "fixed" whatever that might mean, but to regard it as a
"quantity" is more than questionable.

Set theory depends on actual infinity.

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

Bijection means completeness.

No.  Bijection just means a 1-1 correspondence between the elements of
two sets.  Nowhere in that definition is any mention of completeness.
The bijection can be just as easily a "potential" bijection as an
"actual" bijection.

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. In actual infinity there are more natural numbers than even
natural numbers.

Depending on how you pair them, there can also be more even natural
numbers than natural numbers.  The idea of one countable set being
"bigger" than another countable set is simply nonsense.

[ .... ]

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

No, that is wrong.

Feel free to give an example of a "bijection" between "potentially
infinite" sets which are countable, which fails to meet the definition of
bijection as I have outlined above.

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.

The diagonal number _is_ complete.  Remember, we are not constructing it
tiringly one digit per second, or anything like that.  The entire number
is defined and simply exists.  There is no time in set theory.

Impossible. All that is nonsense.

The nonsense is the outdated notion of "potential" and "actual" infinity.
Mathematics no longer has these.  All your arguments demonstrate why
these terms are incoherent.  "Infinite" stands on its own.

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.

Again, there is no time in set theory.  Given a purported complete list
of real numbers, the diagonal number simply exists, and is not in that
list.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).