Path: ...!weretis.net!feeder9.news.weretis.net!news.nk.ca!rocksolid2!i2pn2.org!.POSTED!not-for-mail
From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Tue, 10 Dec 2024 07:17:20 -0500
Organization: i2pn2 (i2pn.org)
Message-ID: <5805ad50ebff3400d1370d8c99790cbc727a340a@i2pn2.org>
References: <vg7cp8$9jka$1@dont-email.me>
 <50c82b03-8aa1-492c-9af3-4cf2673d6516@att.net> <vip5mo$p0da$1@dont-email.me>
 <vipb6l$qfig$1@dont-email.me> <viplj0$t1f8$1@dont-email.me>
 <5a122d22-2b21-4d65-9f5b-4f226eebf9d4@att.net> <viq3i2$105iq$1@dont-email.me>
 <e055ec41-a98d-4917-802f-169575a5b556@att.net> <virq3t$1gs07$1@dont-email.me>
 <c8faf784-348a-42e9-a784-b2337f4e8160@att.net>
 <3af23566-0dfc-4001-b19b-96e5d4110fee@tha.de>
 <ae606e53-0ded-4101-9685-fa33c9a35cb9@att.net> <viuc2a$27gm1$1@dont-email.me>
 <8a53c5d4-4afd-4f25-b1da-30d57e7fe91c@att.net> <vj1acu$31atn$3@dont-email.me>
 <ec451cd6-16ba-463d-8658-8588093e1696@att.net> <vj2f61$3b1no$1@dont-email.me>
 <10ebeeea-6712-4544-870b-92803ee1e398@att.net> <vj3tl0$3nktg$2@dont-email.me>
 <1f1a4089-dfeb-45f8-9c48-a36f6a4688fb@att.net> <vj6bqo$b6bt$1@dont-email.me>
 <b09445be167b757878741be04c87cf76d24d9786@i2pn2.org>
 <vj6psc$dp01$1@dont-email.me>
 <84818a4f5d3795b746b017ad0861a3d818c5b053@i2pn2.org>
 <vj8vd0$stav$1@dont-email.me>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Tue, 10 Dec 2024 12:17:20 -0000 (UTC)
Injection-Info: i2pn2.org;
	logging-data="2178253"; mail-complaints-to="usenet@i2pn2.org";
	posting-account="diqKR1lalukngNWEqoq9/uFtbkm5U+w3w6FQ0yesrXg";
User-Agent: Mozilla Thunderbird
In-Reply-To: <vj8vd0$stav$1@dont-email.me>
Content-Language: en-US
X-Spam-Checker-Version: SpamAssassin 4.0.0
Bytes: 4510
Lines: 74

On 12/10/24 3:50 AM, WM wrote:
> On 10.12.2024 01:45, Richard Damon wrote:
>> On 12/9/24 8:04 AM, WM wrote:
>>> On 09.12.2024 13:03, Richard Damon wrote:
>>>> On 12/9/24 4:04 AM, WM wrote:
>>>>> On 08.12.2024 19:01, Jim Burns wrote:
>>>>>
>>>>>> You (WM) are considering
>>>>>> infinite dark.finite.cardinals,
>>>>>> which do not exist.
>>>>>>
>>>>> Then analysis is contradicted in set theory.
>>>>> ∀n ∈ ℕ: E(1)∩E(2)∩...∩E(n) = E(n).
>>>>> The limit of the left-hand side is empty, the limit of the right- 
>>>>> hand side is full, i.e. not empty.
>>>>> I do not tolerate that.
>>>>>
>>>> By your logic, 1 equals 0, 
>>>
>>> No, that are two different sequences.
> 
>> But since both 0^x and x^0 as x approaches 0 approach 0^0, your logic 
>> says that 0^0 is both 0 and 1.
> 
> You should check your "logic". When two different sequences have 
> different limits, this does not mean that the limits are identical. By 
> the way 0^0 = 1 is simply a definition.

But that is just showing the error in YOUR logic.

The set of 0^x for all non-zero x is clearly 0, so since 0^0 is clearly 
the result of taking 0^x to the limit of x -> 0, that says, by your 
logic, that 0^0 must be 0.

The ser of x^0, for all non-zero x is clearly 1, so since 0^0 is clearly 
the result of taking 1^x to the limit of x -> 1, that says, by your 
logic, that 0^0 must be 1.

Thus 0^0 is both 0 and 1, but it must have just a value, so 0 must be 1.

(We can make it a limit to infinity, by replacing x with 1/n)

That is the error of assuming that just because a sequence apporaches a 
value as a limit (like your 1/10, 1/10, 1/10 ...) that this must be the 
"value" of the item at that limit.


>>
>> Just because you have a sequence, doesn't mean you can talk about the 
>> end infinite state at the "end" of the sequence.
> 
> The end infinite state is a set.

The end state is an infinite set, that doesn't have the same properties 
of the finite sets the form the sequence.

>>
>>  you have two sequences that seem to go to the same infinte set at the 
>> end,
> 
> Two sequences that are identical term by term cannot have different 
> limits. 0^x and x^0 are different term by term.

Which isn't the part I am talking of, it is that just because each step 
of a sequence has a value, doesn't mean the thing that is at that limit, 
has the same value.

This ERROR seems to be a fondation of tour logic, a rule that tends to 
work in the finite realm, but fails at many infinities.

> 
> REgards, WM
> 
>