Re: Replacement of Cardinality

Am 05.08.2024 um 01:41 schrieb Jim Burns:

On 8/4/2024 7:12 PM, Moebius wrote:

Am 04.08.2024 um 22:16 schrieb Jim Burns:

On 8/4/2024 2:13 PM, WM wrote:

 

When all x are involved,
the universal quantifier is usually not written.

>

When a universal quantifier is not written,
it is implicit, and
it can only stand implicitly outside the formula.

>
Right.
Still there's a distinct problem with "implicit quantification"
IN THIS CASE.
(Hence WM's "argument" is nonsense anyway.)

 Yes.
 

Here we need "∀x > 0".
Clearly an "implicit quantifier" does not know that he's restricted to "x > 0".

 We can re.write
∀ᴿx>0: NUFᵈᵉᶠ(x) = ℵ₀

Let's (for simplicity) assume that our domain of discourse is IR:
∀x>0: NUFᵈᵉᶠ(x) = ℵ₀
Then we can re.write
∀x>0: NUFᵈᵉᶠ(x) = ℵ₀

as
∀x: x > 0 ⇒ NUFᵈᵉᶠ(x) = ℵ₀
and, implicitly quantified, as
x > 0 ⇒ NUFᵈᵉᶠ(x) = ℵ₀

Sure, but WM's "implicitely quantified" formula did not consist of an implication with antecedence "x > 0".
To make a long story short: "implicit quantification" does not work for (with?) "restricted quantifiers".
Again: I may claim that
        NUFᵈᵉᶠ(x) = ℵ₀
is "implicitely (all)quantified".
But
        ∀x: NUFᵈᵉᶠ(x) = ℵ₀
would be wrong.
Nuff said.