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

On 10/30/2024 06:04 PM, Ross Finlayson wrote:

On 10/30/2024 05:14 PM, Jim Burns wrote:

On 10/30/2024 3:55 PM, WM wrote:

On 30.10.2024 18:05, Jim Burns wrote:

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∀ᴿx > 0: ∀n ∈ ℕ: ⅟⌈n+⅟x⌉ ∈ ⅟ℕ∩(0,x]
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∀ᴿx > 0: NUF(x) = |ℕ|

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Is ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 wrong?

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Neither
  ∀n ∈ ℕ: 1/n - 1/(n+1) > 0
nor
  ∀ᴿx > 0: ∀n ∈ ℕ: ⅟⌈n+⅟x⌉ ∈ ⅟ℕ∩(0,x]
is wrong.
>
>

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That's Archimedean alright.
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These days perhaps you've heard of non-Archimedean,
and not only useless is well-meaning yet saying-nothing
extensions yet actually non-standard countable and
this kind of thing, where of course it's so that
Archimedean usually enough means super-Archimedean.
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Archimedes would be proud.
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Then here for example there's that Aristotle identifies
the concerns of Eudoxus, infinite-divisibility, and
those of Democritus, atomism, then for what's Aristotlean
goes the way of Eudoxus if merely for not being vacuous,
yet the super-Aristotlean is where's the super-Archimedean.
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Here thusly it's: "infinitary".
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Then, for how to explain Nyquist and Shannon and signal
theory and information theory, and, Fourier series the
derivation, and, Dirichlet function, I'm not quite sure
where in the classical that is, super-classical that being.
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Yet, the derivations what arrive at it are as of the
sorts of double-reductio or reductio-reductio, according
to the language of limits and all things equal, and
particularly the completions.
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To refer to a "scalar infinity" as "cardinality of naturals"
is a sort of foolish thing, because cardinality is _counts_
and the scalar is _numbering_. The counting's usually enough
matching according to a schema, the other's issuing unique
identifiers. While that is so, it's also so that there's a
usual consideration about inductive arguments in the limit
that's forgotten or never understood, to the effect that
not only does there exist a larger input closer, that there
exists a much, much larger input that's no distant, the limit.
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The infinite limit, ....
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Use the language of infinite limit, then all inductive followers
have nothing to do but follow, then though as with regards to
things like counterexamples in convergence and so on, and
the non-standard, that's for deductive analysis in the greater
space, sometimes these days attributed to Ramsey or Birkhoff.
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Or me, ....
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Here for example is a recent paper about these "meeting in the
middle" type things, like "Borel versus Combinatorics".
https://arxiv.org/abs/2203.10396
"In this paper, we show that the natural notion of largeness
given by upper density reveals that these phenomena meet in the
countable ..."
"Our methods explore a connection with the notion of convergence
in the theory of limits of dense combinatorial objects, ..."
See, in Cantor space, all the sequences of 0's and 1's, there's
that Borel has that "almost all" are one way, and combinatorics
has "almost all" are the other way, and thus, it's established
there are law(s) of large numbers independent standard models
of integers either way, and furthermore: one in the middle.
Independent standard naive number theory, ..., if there is one at all.