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.