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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Wed, 11 Dec 2024 22:53:24 +0100
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On 11.12.2024 21:58, Jim Burns wrote:
> On 12/11/2024 2:57 PM, WM wrote:
>> On 11.12.2024 20:27, Jim Burns wrote:
> 
>>> ⋂{E(i):i} = {}.
>>
>> Of course. But
>> all intersections with finite contents
>> are invisible.
> 
> We know about what's invisible by
> assembling finite sequences holding only
> claims which are true.or.not.first.false.
> 
> We know that
> each claim in the claim.sequence is true
> by _looking at the claims_
> independently of _looking at the invisible_
> 
> _It doesn't matter_
> whether any finite.cardinals are invisible.
> Each finite cardinal is finite, and
> that is enough to start
> a finite sequence of claims holding only
> claims which are true.or.not.first.false
> -- claims about each finite.cardinal, visible or not.
> 
> Some claims seem too dull to need verifying.
> "Is a finite.cardinal finite?"
> Better to ask "Is the Pope Catholic?"
> But such obviously.true claims start us off.
> 
> Other claims, the more interesting claims,
> can be verified as not.first.false
> _by looking at the claims_
> NOT by looking at finite.cardinals
> Look at q in ⟨p p⇒q q⟩
> There is no way in which q can be first.false.
> It doesn't matter what q means, or what p means.
> We can see q is not.first.false in that sequence.
> 
> Repeat the pattern ⟨p p⇒q q⟩ and a few others
> for a whole finite sequence of claims,
> and
> that whole finite sequences of claims
> holds no first false claim,
> and thus holds no false claim.
> 
> Which we know by _looking at the claims_
> 
>>> Therefore,
>>> one.element.emptier ℕ\{0}
>>> is not.smaller.than ℕ
>>
>> It is a smaller set.
> 
> For each k in ℕ
> there is unique k+1 in ℕ\{0}
> 
>> Cardinalities are not useful.
> 
> And yet, by ignoring them,
> you (WM) end up wrong about
> ⎛ For each k in ℕ
> ⎝ there is unique k+1 in ℕ\{0}
> 
All that waffle only in order to avoid the crucial question?
Simply answer by yes or no: Is the complete removal of natural numbers 
from the sequence of intersections bound by the law
∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
or not?

Regards, WM