Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 01/08/2025 07:12 AM, Alan Mackenzie wrote:

Ross Finlayson <ross.a.finlayson@gmail.com> wrote:

On 01/07/2025 03:36 AM, Alan Mackenzie wrote:

Moebius <invalid@example.invalid> wrote:

Am 05.01.2025 um 12:28 schrieb Alan Mackenzie:

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None of the above extracts is about mathematics.  If there were such
things as "potential" and "actual" infinity in maths, then they would
make a difference to some mathematical result.  There would be some
theorem provable given the existence of PI and AI which would be false
or unprovable with just plain infinite, or vice versa.  Or something
like that.  Nobody in this discussion has so far attempted to cite
such a result.

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When I did my maths degree, several decades ago, "potential
infinity" and "actual infinity" didn't get a look in.  They weren't
mentioned a single time.  Instead, precise definitions were given to
"finite" and "infinite", and we learnt how to use these definitions
and what could be done with them.

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Sure. But this needs a context; usually (some sort of) set theory.

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The only people who talk about "potential" and "actual" infinity are
non-mathematicians who lack understanding

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Like those mentioned above?

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OK, and mathematicians in their time off.  ;-)

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Oh, how about yin-yang ad-infinitum,
a simple and graphical example of
something that is, in the infinite,
not what it is, in the limit,
that it's actual infinite limit,
differs its potential infinite limit.

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What do you mean by the "yin-yang ad-infinitum", exactly?
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I'm not even sure how you would define, or distinguish "actual infinite
limit" and "potential infinite limit".  Do such things already have
accepted mathematical definitions?
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It's also well known that a limit frequently has properties not in any of
the terms.  For example, the limit of a sequence of rational numbers is
often irrational.  Is this what you meant?
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Hey, thanks for asking.
So, the ratio of circumference to diameter,
of a circle, is pi. The area of a pi is pi d,
d for diameter, p for perimeter, say, about
the isoperimetric theorem and perfect circles.
Would you agree that there is such a thing as
a perfect circle? It's a usual concept associated
with mathematical platonism.
So, imagine the diameter's vertical, and,
draw two non-concentric circles that are
incident each other and the outer, two
inner circles. So, the sum of their diameters
is d, their diameter each is d/2, their circumference
each is pi d/2, and that times two equals pi d,
the same as the outer circle.
Then, that sort of looks like a yin-yang,
bending that around, yet that's just for imagery.
So, repeat this process, and, inductively,
the sum of all the inner circles is pi d,
yet, in the limit, the areas of the entire
outer circle are removed, resulting a flat
line. So, in the infinite limit, it is
different than all its approximations
or what according to induction never changes.
So, by employing some features of the isoperimetric
theorems with regards to circles halving their diameters,
there's a quick example of something that according
to inductive inference has a limit, and it's a constant,
yet in the infinite limit, equals d or d/2, not pi d.
So, it's not just about least-upper-bound property,
and sequences with the Cauchy property that
are algebraic yet not rational, or transcendental,
it's actually about that inductive inference itself
results itself not correct.
And, it demonstrates that a deductive inference result,
arrives at a construction, that in the "actual infinite
limit", i.e. the completion of what's limited,
is more than merely "potential".
As you requested, ....