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

It happens that Chris M. Thomasson formulated :

On 10/8/2024 2:21 PM, Richard Damon wrote:

On 10/8/24 4:17 PM, Chris M. Thomasson wrote:

On 10/8/2024 7:10 AM, Richard Damon wrote:

On 10/8/24 10:00 AM, Alan Mackenzie wrote:

Richard Damon <richard@damon-family.org> wrote:

On 10/8/24 9:11 AM, Alan Mackenzie wrote:

Richard Damon <richard@damon-family.org> wrote:

On 10/8/24 5:42 AM, Alan Mackenzie wrote:

Richard Damon <richard@damon-family.org> wrote:

On 10/7/24 7:44 AM, Alan Mackenzie wrote:

Richard Damon <richard@damon-family.org> wrote:

On 10/7/24 7:13 AM, joes wrote:

Am Mon, 07 Oct 2024 10:13:21 +0200 schrieb WM:

On 06.10.2024 17:55, joes wrote:

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:

>

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).

Actual means all, but not more. This implies a last before ω.
The infinity means an end cannot be determined. It is produced
by the dark numbers.

Actually infinite means infinite, which doesn’t change when you
add or subtract a finite number.

>
>

Actual infinity doesn't exist for us finite beings.

>

English language tip: The "Actually" in that sentence was not
attached to the word "infinite", it meant something like "This is
really true:".

>

But all his reference to the word "Actually" are part of his trying to
define the term "Actual Infinity".

>

OK, maybe you're right, there.  The semantics are a bit ambiguous.
Joes is not a native English speaker.  Apologies to Joes.

>

Anyhow, what do you mean when you say that "actual infinity
doesn't exist"?  I think we established over the weekend that for
a mathematical entity not to exist, it must cause a contradiction.
Or something like that.

>

So what contradiction would the existence of actual infinity cause?

>

It implies that there exists a first positive real, rational number or
unit fraction for one (at least the way WM uses it).

>

Whoa!  There're rather a lot of argument steps missing there.  Just
because WM asserts the existence of both actual infinity and a first
strictly positive unit fraction doesn't mean the one implies the other.

>
>

It does in his logic, which is all that matters to him.

>

Were we talking about WM's "logic", just there?  I don't think I was.

>

But my quote came from his logic.

>
Ah, OK.
>

Yes, it is a wrong conclusion, but that error is based on his initial
assumption that something could be used that isn't available as a
understandable entity to us finite beings.

>

I think infinity is understandable.  I think I understand it.  My
position is that the distinction between "potential infinity" and "actual
infinity" is bogus.  It makes no difference in mathematics, which is
probably why the terms have vanished from mathematical discourse.

>

The difference is that in his "actual infinity" the generation process
is complete and nothing can change.

>
In a sense, the "actual infinity" _is_ the generation process, which is
fixed and complete.

>
But only after that infinite process has completed. Finite beings can't actually SEE that result.[...]

>
When does an infinite process finally complete? If it does, well, it was NOT infinite in any way shape, or form... Right?

>
The infinite process completes in the infinite, thus beyond what we can perceive, but only dimly imagine.
>
That is why we can't have actual infinity, we can't get to the infinite to see it done.

>
since a step-by-step process for the naturals aka:
>
1, 1+1, 1+1+1, ...
>
Will never end, well... That does not mean there is a largest natural. WM is strange on this aspect. See, this right here is pondering on the infinite from a finite being... ;^)

 I suggest forgetting the whole 'finite being' idea. Mathematics is filled with abstractions of things that are not possible in real life and time. It is not necessary to build the natural numbers step by step in sequence in an impossible supertask. It is an object in mathematics, boom there it is. :)