Re: Replacement of Cardinality (quantifier disambiguation)

On 08/04/2024 05:49 PM, Ross Finlayson wrote:

On 08/04/2024 05:24 PM, Moebius wrote:

Am 05.08.2024 um 01:07 schrieb Chris M. Thomasson:

On 8/4/2024 8:35 AM, WM wrote:

Le 04/08/2024 à 02:15, Moebius a écrit :

Am 03.08.2024 um 21:54 schrieb Jim Burns:

On 8/3/2024 10:23 AM, WM wrote:

>

NUF(x) = ℵ₀ for all x > 0 is wrong.

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Nonsense.
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Actually, Ax > 0: NUF(x) = ℵ₀.

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You mean that there are ℵ₀ unit fractions smaller than all positive x?

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Obviously not.
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What I mean is that for all positive x there are ℵ₀ unit fractions
smaller than x.
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Impossible. [...] Not even one unit fraction can be smaller than all
positive x.

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No one (except WM) claimed that there's a unit fraction which is smaller
than all positive x.
>

Huh?

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WM is constantly mixing up
>
          ∀x > 0: ∃^ℵ₀ u ∈ ⅟ℕ: u < x   (true)
with
          ∃^ℵ₀ u ∈ ⅟ℕ: ∀x > 0: u < x   (false) .
>
(Here ⅟ℕ = {1/n : n e IN} is the set of all unit fractions.)
>

Say x = 1/2, there are infinite smaller unit fractions, say, 1/4, 1/5,
1/6, ect... However there is only one larger one, 1/1. See? No
smallest one for 1/0 is not a unit fraction! There is a largest one,
1/1...
>
They tend to zero, but there is no smallest one...

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Yeah.
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Proof: If s is a unit fraction then 1/(1/s + 1) is a unit fraction which
is smaller than s (for each and every s).
>

See?

>
>

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See, you meant the same things but were using different modes
for each/any/every/all and now you think each other were wrong.
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Or, you know, cleared that up.
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There are different modes of universal quantification and
they're particularly relevant in arguments about them.
>
>
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(Also "material implication" is "quasi-modal".)
>
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In front of any term are infinitely many implicit quantifiers,
the term being infinitely many implicit variables, that
a usual modest "for-any x" is "to the omega".
>
Here though specifically "any" and "each" don't mean
the same thing as "every" and "all".
>
Though, the usual "universal quantifier" doesn't know that.
>
>

Natural language can convey a lot, and sometimes
it's more relevant to make a "generous" reading and
figure that someone meant something correct when
it could be read as wrong, vis-a-vis a usual sort of
"robot's first syntax exception: bzzt... error... error...,
does not compute", instead of, "... can not compute".
It's similar when dealing with cranktrolls of the various
sorts often of the sockpuppet variety like "dumb WM"
and "AP the lesser", with there being of course an "AP the greater",
about making a generous reading, about what _can_ be
salvaged from their argument, deconstructing to what's
not discarded, instead of just flat balking and catching
just the diagnosis of the error, not the success.
So, when Dumbem (you know, "dumb-'em down") has
a, ..., "half-truth" of sorts, there's still keeping that what's
true about things, besides knowing what's wrong with it,
and offering particularly how to go about so that a less
generous reading would have nothing to complain about,
providing correction not just rejection.
Here for example the "number theory says half of the integers
are even" can help somebody when they're told "cardinality
the size relation is about sets and doesn't know the modularity
of numbers per se though of course you could also just contrive
sets of functions by blowing up and taking back down the modular
so that cardinality as a model of size relation can have models
made for it of any other size relation like density or for that
matter OUTPACING", and these kinds of things, when that a
tool like the inequality transitively of cardinals makes itself
so absolute, it's just re-used because then it must be possible
to build such matters of relation in terms of it, establishing
the extensionality either way, in the numbers' terms or the
sets' terms.
Then though that infinite completions either complete
or not, there's a pretty strong argument that they either
do or don't, that if they do they do and if they don't they don't.
This is for something like Zeno and the limit and the infinite limit,
there being a difference, and that Zeno particularly says that
"if you don't get all the way across, then close enough is also
close enough to half, to a quarter, and on down the inverse
powers of two, to none".
I.e. Zeno explains that the analytical bridge has an inductive
impasse and must be surpassed as an infinite super-task.