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

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.

--
Alan Mackenzie (Nuremberg, Germany).