Path: ...!Xl.tags.giganews.com!local-4.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail
NNTP-Posting-Date: Sun, 29 Dec 2024 04:15:59 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary, effectively)
Newsgroups: sci.math
References: <vg7cp8$9jka$1@dont-email.me> <vjufr6$29khr$3@dont-email.me>
 <4bf8a77e-4b2a-471f-9075-0b063098153f@att.net> <vjv6uv$2dra0$1@dont-email.me>
 <31180d7e-1c2b-4e2b-b8d6-e3e62f05da43@att.net> <vk1brk$2srss$7@dont-email.me>
 <bb80c6c5-04c0-4e2d-bb21-ac51aab9e252@att.net> <vk23m7$31l8v$1@dont-email.me>
 <bce1b27d-170c-4385-8938-36805c983c49@att.net> <vk693m$f52$2@dont-email.me>
 <a17eb8b6-7d11-4c59-b98c-b4d5de8358ca@att.net> <vk7dmb$7mh2$2@dont-email.me>
 <b72490c1-e61a-4c23-a3a5-f624b2c084e4@att.net> <vk8tbq$j9h1$1@dont-email.me>
 <bd7dfdc7-6471-4fe6-b078-0ca739031580@att.net> <vklumc$3htmt$1@dont-email.me>
 <c03cf79d-0572-4b19-ad92-a0d12df53db9@att.net>
 <n9CdnR02SsevtPL6nZ2dnZfqnPidnZ2d@giganews.com>
 <45a632ed-26cc-4730-a8dd-1e504d6df549@att.net> <vkpa98$dofu$2@dont-email.me>
 <15f183ae29abb8c09c0915ee3c8355634636da31@i2pn2.org>
 <UY-cndwXAt7-4O36nZ2dnZfqnPYAAAAA@giganews.com>
 <cc538b04-66c2-453e-8abf-e1a425cc2b77@att.net>
 <loKcnQ1ho8WXVe36nZ2dnZfqn_udnZ2d@giganews.com>
From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sat, 28 Dec 2024 20:15:28 -0800
User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101
 Thunderbird/38.6.0
MIME-Version: 1.0
In-Reply-To: <loKcnQ1ho8WXVe36nZ2dnZfqn_udnZ2d@giganews.com>
Content-Type: text/plain; charset=utf-8; format=flowed
Content-Transfer-Encoding: 8bit
Message-ID: <dbKdnTjaMa1iUe36nZ2dnZfqnPWdnZ2d@giganews.com>
Lines: 108
X-Usenet-Provider: http://www.giganews.com
X-Trace: sv3-iR1AlPyzuJBaCrIxTB7hBWSpJZy49Ug/Yjs8+DTgkNQU1rPPXnJIsabN+quCGigCz4P7FInnYgcX3cQ!ZitIJK2CxnHmyNBT0krar8DIx0xzNaHiaUwjJ385ZzticB/8ie+D2bTOqDNpMwuRLfELlPA+uJfR
X-Complaints-To: abuse@giganews.com
X-DMCA-Notifications: http://www.giganews.com/info/dmca.html
X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers
X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly
X-Postfilter: 1.3.40
Bytes: 5263

On 12/28/2024 07:54 PM, Ross Finlayson wrote:
> On 12/28/2024 04:22 PM, Jim Burns wrote:
>> On 12/28/2024 5:36 PM, Ross Finlayson wrote:
>>> On 12/28/2024 11:17 AM, Richard Damon wrote:
>>
>>>> [...]
>>>
>>> Consider
>>> a random uniform distribution of natural integers,
>>> same probability of each integer.
>>
>> A probability.measure maps events
>> (such as: the selection of an integer from set S)
>> to numbers in real.interval [0,1]
>>
>> For each x ∈ (0,1]
>> there is a finite integer nₓ: 0 < ⅟nₓ < x
>>
>> ⎛ Assume a uniform probability.measure
>> ⎜ P: 𝒫(ℕ) -> [0,1]
>> ⎜ on ℕ the non.negative integers.
>> ⎜ ∀j,k ∈ ℕ: P{j} = P{k} = x
>> ⎜ x ∈ [0,1]
>> ⎜
>> ⎜ If x = 0, Pℕ = ∑ᵢ₌᳹₀P{i} = 0 ≠ 1
>> ⎜ P isn't a probability.measure.
>> ⎜
>> ⎜ If x > 0,  0 < ⅟nₓ < x
>> ⎜ P{0,…,nₓ+1} > (nₓ+1)/nₓ > 1
>> ⎝ P isn't a probability.measure.
>>
>> Therefore,
>> there is no uniform probability.measure on
>> the non.negative integers.
>>
>>> Now, you might aver
>>> "that can't exist, because it would be
>>> non-standard or not-a-real-function".
>>
>> I would prefer to say
>> "it isn't what it's describe to be,
>> because what's described is self.contradictory".
>>
>>> Then it's like
>>> "no, it's distribution is non-standard,
>>> not-a-real-function,
>>> with real-analytical-character".
>>
>> Which is to say,
>> "no, it isn't what it's described to be"
>>
>>
>
> You already accept that the "natural/unit
> equivalency function" has range with
> _constant monotone strictly increasing_
> has _constant_ differences, _constant_,
> that as a cumulative function, for a
> distribution, has that relating to
> the naturals, as uniform.
>
> And that they always add up to 1, ....
>
>
> That most certainly is among the definitions
> of a distribution, the probabilities even
> range between 0 and 1.
>
> So, even though you refuse that this is
> a real function, because it's not, yet
> it's also a distribution, which it is.
>
> "Standardly as a limit of functions"
> if you won't, like Dirac's unit impulse
> function, never actually so esxcept its completion,
> yet necessarily actually so in the derivations
> that depend on it, like Fourier-style analysis,
> "real analytical character", say.
>
> Then also that it really is "a continuous
> domain" and "a discrete distribution",
> just keeps pointing out how special it is,
> "the natural/unit equivalency function".
>
>
> One of at least three set-theoretic accounted
> models of continuous domains, and establishing
> that there are non-Cartesian functions via
> an anti-anti-diagonal-argument.
>
>

So that ZF isn't internally inconsistent
with respect to real numbers, ....

.... and continuous domains.


Of course one could always just write it
in some otherwise equi-interpretable
"theory of one relation" yet it's so
that ZF and ZFC are the most well-known
and well-explored axiomatic set theories.


"Pick either: get both."