Re: Incompleteness of Cantor's enumeration of the rational numbers (research)

On 11/13/2024 06:04 PM, Ross Finlayson wrote:

On 11/13/2024 05:43 PM, Jim Burns wrote:

On 11/13/2024 7:05 PM, FromTheRafters wrote:

Jim Burns formulated on Wednesday :

On 11/13/2024 4:29 PM, WM wrote:

On 13.11.2024 20:38, Jim Burns wrote:

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----

 Bob.

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KING BOB!
https://www.youtube.com/watch?v=TjAg-8qqR3g
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If,
  in a set A which
  can match one of its proper subsets B,

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That is nonsense too.

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[repaired]
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A finite sequence of claims in which
each claim is true.or.not.first.false
is
a finite sequence of claims in which
each claim is true.
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Some claims are true and we know it
because
they claim that
when we say this, we mean that,
and we, conscious of our own minds, know that
when we say this, we mean that.
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Some claims are not.first.false and we know it
because
we can see that
no assignment of truth.values exists
in which they are first.false.
q is not first.false in ⟨ p p⇒q q ⟩.
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Some finite sequences of claims are
each true.or.not.first.false
and we know it.
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When we know that,
we know each claim is true.
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We know each claim is true, even if
it is a claim physically impossible to check,
like it would be physically impossible
to check each one of infinitely.many.
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We know because
it's not checking the individuals
by which we know.
It's a certain sequence of claims existing
by which we know.
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In my source window:

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[...]

That is nonsense too.

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A finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀
in which
each claim is true.or.not.first.false
is
a finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀
in which
each claim is true.

[...]
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================================================
I follow some of this mostly from context. :)

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Sorry about that.
The other fonts weren't strictly necessary,
I just had a brainstorm over
how to (maybe) explain logical validity better,
and I couldn't resist.
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Some usual laws, or criteria, rather, of convergence,
fail, for example Stirling's formula.
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When are they ever wrong? Are there simply more
than a usual naive law of large numbers what's
merely the law of small numbers?
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Then, asymptotic freedom, or the Arago spot, make
examples of what do not arrive from inductive inference.
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So, these super-classical concerns are a thing.
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There's one rhyme,
"I like traffic lights,
I like traffic lights,
I like traffic lights, ...."
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Also usually called slippery slope,
shifting sands, or ad absurdam.
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Usually of course arrived at ultimately.untrue
from more objective concerns.
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Take a look to Chrysippus, he establishes great
grounds for modal (mood-al) logic and relevance logic about
hundreds of years before Plotinus arrived at
the "material inductive implication" the "quasi-modal",
and provides reasoning for more thorough accounts
when people might not have time to read and follow
both Aristotle's Prior, and Posterior sur-rounds
of inference.
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Or, "not.first.false" must yet also be "not.ultimately.untrue",
when _all_ the cases are run out.
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(Or, maybe it's the other way, ....)
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As long as you might agree that _all_ your stipulations be
read off in any order, that might help, it's a usual
criterion of constructivism.
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For structuralists and not merely the shallow feels.
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A finitary Kronecker's lemma and large deviations in the Strong Law of
Large numbers on Banach spaces
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Duality on symmetric multiple polylogarithms
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Continuity of matings of Kleinian groups and polynomials
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Complexity of Finite Borel Asymptotic Dimension
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On limiting distributions of arithmetic functions
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Products of pseudofinite structures
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Spectral equivalence of nearby Lagrangians
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Sparser Abelian High Dimensional Expanders
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Probability Laws Concerning Zeta Integrals
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-- https://www.arxiv.org/list/math/recent
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Unified analysis of non-Markovian open quantum systems in Gaussian
environment using superoperator formalism
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A generalization of the martingale property of entropy production in
stochastic systems
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Superintegrability and Coulomb-Oscillator Duality
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Central limit theorem for the focusing Φ4-measure in the infinite volume
limit
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Type IIA String Theory and tmf with Level Structure
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Emergent Geometry from Quantum Probability
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-- https://www.arxiv.org/list/math-ph/recent
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Pretty much each of these involves continuum analysis
and non-standard continuum analysis, what with regards
to what foundations _owes_ mathematics of a theory
with continuity (infinitesimals), and laws of large numbers (infinities).
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Of course, which break-down or stop right away not.first.false.
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A sampling of the past week or two, ....
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