Sujet : Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
De : james.g.burns (at) *nospam* att.net (Jim Burns)
Groupes : sci.mathDate : 20. Dec 2024, 19:48:35
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <bce1b27d-170c-4385-8938-36805c983c49@att.net>
References : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
User-Agent : Mozilla Thunderbird
On 12/19/2024 4:37 PM, WM wrote:
On 19.12.2024 19:29, Jim Burns wrote:
On 12/19/2024 9:50 AM, WM wrote
On 18.12.2024 21:21, Jim Burns wrote:
⋂{E(1),E(2),...} is empty.
Other numbers are not in the argument.
>
But E(1) is full.
The only way to get rid of content is
to proceed by
∀k ∈ ℕ:
∩{E(1), E(2), ..., E(k+1)} =
∩{E(1), E(2), ..., E(k)} \ {k},
i.e. to lose one number per term of
the function
f(k) = ∩{E(1), E(2), ..., E(k)}.
>
f(k) = E(k) = ⟦k,ℵ₀⦆
Yes,
after infinitely.many one.number.losses,
infinitely.many numbers are lost.
>
That means all numbers are lost by loss of
one number per term.
>
That implies finite endsegments.
Q. What does 'finite' mean?
----
Set ⟦0,ℵ₀⦆ of finite cardinals.
Fore.segments of the finite.cardinals
are finite sets.
k ∈ ⟦0,ℵ₀⦆ ⇒ #⟦0,k⦆ ∈ ⟦0,ℵ₀⦆
Set ⟦0,ℵ₀⦆ is not a finite set.
⎛ ∀k ∈ ⟦0,ℵ₀⦆:
⎜ #⟦0,ℵ₀⦆ ≥ #⟦0,k+1⦆ > k
⎜ #⟦0,ℵ₀⦆ ≠ k
⎜
⎝ #⟦0,ℵ₀⦆ ∉ ⟦0,ℵ₀⦆
⎛ Q. What does 'finite' mean?
⎜
⎜ An answer which does not describe
⎜ the arithmetic of sheep and pebbles
⎝ is an instance of bait.and.switch.
The union of two finite sets is a finite set.
#A ∈ ⟦0,ℵ₀⦆ ∧ #B ∈ ⟦0,ℵ₀⦆ ⇒ #(A∪B) ∈ ⟦0,ℵ₀⦆
Fore.segment ⟦0,𝔊⦆ is finite.
If end.segment ⟦𝔊,ℵ₀⦆ were finite,
then ⟦0,ℵ₀⦆ = ⟦0,𝔊⦆∪⟦𝔊,ℵ₀⦆ would be finite,
but ⟦0,ℵ₀⦆ is not finite.
Therefore,
end.segment ⟦𝔊,ℵ₀⦆ is not finite.
#⟦0,𝔊⦆ ∈ ⟦0,ℵ₀⦆ ∧ #⟦𝔊,ℵ₀⦆ ∈ ⟦0,ℵ₀⦆ ⇒ #⟦0,ℵ₀⦆ ∈ ⟦0,ℵ₀⦆
#⟦0,𝔊⦆ ∈ ⟦0,ℵ₀⦆ ∧ #⟦0,ℵ₀⦆ ∉ ⟦0,ℵ₀⦆ ⇒ #⟦𝔊,ℵ₀⦆ ∉ ⟦0,ℵ₀⦆
That means all numbers are lost by loss of
one number per term.
>
That implies finite endsegments.
No.
Yes, each number is lost by loss of
one number per term.
However,
each end.segment is not finite.
However,
each number is not.after infinitely.many numbers,
each loss is not.after infinitely.many losses.
>
As long as
any natural number remains,
the sequence of lost numbers is finite
(ended by the remaining number).
Claims P⇒Q and ¬Q⇒¬P are both.true or both.false.
P⇒Q above, ¬Q⇒¬P below.
As long as
the sequence of lost numbers is infinite,
no natural number remains.
I agree.