Re: Replacement of Cardinality (defining numbers as half finite, half infinite)

On 08/06/2024 05:37 PM, Moebius wrote:

Am 07.08.2024 um 02:27 schrieb Jim Burns:
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however things stand with supertasks, [...]

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Here my 2 cents: Some "supertasks" (thought experiment) DO lead*) to a
specific "final result" (from a "logical" point of view), some NOT.
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Thomson's Lamp is of the latter type.
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It's final state is simply not "determined" by all it's "earlier" states.
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*) in a certain sense; usually considering some additional assumptions.

In nature, in the very heart of the nucleus of the atom,
where according to plain mathematics the most concentrated
force of the strong nuclear force that holds the atom's
nucleus together would be, is: "asymptotic freedom".
As a supertask, according to induction, in the concentric
model of the force that is the binding energy that holds
together the very substance of matter itself, in the theory,
is "asymptotic freedom".
This is as was theorized by Salam and Weinberg in the 1970's,
and within a few years for which prizes in physics were awarded.
The geometric series, Zeno's, adds up to one. Of course, in
nature, in the mesoscale or the classical, that's always
perfectly true.
The idea that there are anti-inductive results in the
infinitary, is even a bit stronger than the usual apeiron
or non-results in the infinitary, though, the most usual
combined inductive and deductive results, about the
infinitely-divisible in a space, are most strong.
That is to say, calculus is imperfect and only at best
a close approximation and always with a non-zero error term,
except in the infinite limit, where it is perfect, and,
it not only goes to that once, it must be that it goes to
that all the many times, because we don't merely have a
fundamental theorem of differentiation, yet a fundamental
theorem of integration, which sums no less than infinitely
many areas to be correct in the usual sense, and no less.
These days some peopls don't even know that various
conjectures of Goldbach in the number-theoretic, or
about the lengths of arithmetic progressions or Szemeredi,
are rather formally undecide-able and with models of
standard and non-standard integers, both fragments
and extensions, both so and not so.
Then, induction, alone, does not suffice, and,
deduction, together with properties of the space
like constancy in measure and proportion in relation,
make so that the geometric series has a sum.
Calculus is only correct in the _infinite_ limit.
Then, for mathematical platonists, that must be real
somehow, just like there's no perfect circle, yet there
is, an object of mathematics, there's infinity,
an object of mathematics.
Then, results in the non-standard like summing the
geometric series or Zeno's, or everyone's favorite
first non-standard not-a-real-function Dirac's delta,
which you can notice on the Wiki is called a "function",
again, continue.