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

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.

The problem is then that the objects are "fixed", The problem here is
that our concepts of such don't really handle "infinite" objects, we can
think about them "going on to infinity" and it sort of fades out of view
in the distance, but since we have never really sensed "infinity" we
have no way to actually fully understand it.

There are lots of things we haven't really sensed that we can understand.
Electromagnetic waves outside the visible range, speeds where special
relativity is significant, black holes, ....  I don't see that infinity
is all that different.

When we understand the "potential infinity" we see that it goes on
forever, and don't feel a need to get there to see it, since we know we
can't.

In presuming it is fixed, he presumes he can get to that infinite point,
because everything he knows is finite (just like all we have actually
seen is finite) and his logic is based on that intuition.

Yes.

WM's logic seems to be based on the presumption that:

If actual infinity exists ... (with the assumption that if it exists, it
follows the logic we know)

Since such a thing does NOT exist, at least as far as our logic can
handle, NONE of his conclusions apply, since the logic was based on a
untrue premise.

His conclusions largely don't apply whether or not "actual infinity"
exists.  It is his deductive steps which are at fault rather than his
axioms.

In WM's mind, anything that is "fixed" has ends, but an infinite thing
doesn't have all those ends.

Indeed not.

--
Alan Mackenzie (Nuremberg, Germany).