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

Am Sun, 06 Oct 2024 17:26:07 +0200 schrieb WM:

On 06.10.2024 16:52, Alan Mackenzie wrote:

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

 

"A potential infinity is a quantity which is finite but indefinitely
large. For instance, when we enumerate the natural numbers as 0, 1, 2,
..., n, n+1, ..., the enumeration is finite at any point in time, ....

That is a mistake.  The ellipses indicate the enumeration.  There is no
time.  If one must consider time, then the enumeration happens
instantaneously.

In potential infinity there is time or at least a sequence of steps.

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.
10*0.999...999 = 9.99...9990 = 9 + 0.99...9990 < 9 + 0.999...999 ==>
9*0.999...999 < 9 as it should be.

In actual infinity, there is no last 9 (that would not be infinite).
One cannot place something after an infinity; your „last” digits would
have places ω+k. Instead, there are all(!) |N|=Aleph_0 places, meaning
the nines don’t end. How can you shift what you call dark?
Consider the bijection f(x)=x-1 from N\{0} to N.

The above is all very poetic, this supposed difference between "actual"
and "potential" infinite, but it is not mathematical.  There are no
mathematical theorems which depend for their theoremhood on the
supposed distinction between "actual" and "potential" infinite.

Set theory depends on actual infinity. Bijections must be complete. 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.

Your argument is not particular to Canter, you merely picked him to
elevate yourself. Your only problem is not grasping the actual
infinity of the naturals.

--
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.