Re: Replacement of Cardinality (quantifier disambiguation)

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.

>
Nonsense.
>
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.
>
What I mean is that for all positive x there are ℵ₀ unit fractions
smaller than x.
>

Impossible. [...] Not even one unit fraction can be smaller than all
positive x.

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

Huh?

>
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...

>
Yeah.
>
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?

>
>

See, you meant the same things but were using different modes
for each/any/every/all and now you think each other were wrong.
Or, you know, cleared that up.
There are different modes of universal quantification and
they're particularly relevant in arguments about them.
(Also "material implication" is "quasi-modal".)
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.