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From: Moebius <invalid@example.invalid>
Newsgroups: sci.logic,sci.math
Subject: Re: Replacement of Cardinality
Date: Mon, 5 Aug 2024 02:03:03 +0200
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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.