Path: ...!Xl.tags.giganews.com!local-1.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail
NNTP-Posting-Date: Sun, 28 Jul 2024 23:25:13 +0000
Subject: Re: Replacement of Cardinality
Newsgroups: sci.logic,sci.math
References: <hsRF8g6ZiIZRPFaWbZaL2jR1IiU@jntp>
 <881fc1a1-2e55-4f13-8beb-94d1f941b5af@att.net>
 <vg44QVKbPSR4U0Tq71L-fg5yqgM@jntp>
 <85194aeb-1b24-4486-8bcc-4dcd43b4fd2f@att.net>
From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sun, 28 Jul 2024 16:25:31 -0700
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: <85194aeb-1b24-4486-8bcc-4dcd43b4fd2f@att.net>
Content-Type: text/plain; charset=utf-8; format=flowed
Content-Transfer-Encoding: 8bit
Message-ID: <HVudnVg62uHETjv7nZ2dnZfqn_ednZ2d@giganews.com>
Lines: 107
X-Usenet-Provider: http://www.giganews.com
X-Trace: sv3-Q3Ly0dJe9qHtVEMhEDF6S/bDFXB1m6Sw5XgfVkJKeJe8OKqWm7+g6SqKYepxUC2nClkNiRh/Yu9zkHG!3HtjZWaitaQ1x7sJJ2hhOhq4I5VLFhVZ0HUEhRj400AlxrxpPv8/ZMOuE0C8f5YRfyfTFs5/Ci8=
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: 4258

On 07/28/2024 11:17 AM, Jim Burns wrote:
> On 7/28/2024 8:17 AM, WM wrote:
>> Le 27/07/2024 à 19:34, Jim Burns a écrit :
>>> On 7/26/2024 12:31 PM, WM wrote:
>
>>>> _The rule of subset_ proves that
>>>> every proper subset has less elements than its superset.
>
>>> If ℕ has fewer elements than ℕ∪{ℕ}
>>> then
>>> |ℕ| ∈ ℕ
>>
>> |ℕ| = ω-1 ∈ ℕ
>
> ⎛ Each non.{}.set A of ordinals holds min.A
> ⎜
> ⎜ Ordinal j = {i:i<j} set of ordinals before j
> ⎜
> ⎜ Finite ordinal j has fewer elements than j∪{j}
> ⎜
> ⎝ ℕⁿᵒᵗᐧᵂᴹ is the set of ALL finite ordinals.
>
> No finite.ordinal is last.finite,
> no visibleᵂᴹ finite.ordinal,
> no darkᵂᴹ finite.ordinal.
> In particular, no finite.ordinal is ω-1
>
> Also, no before.first infinite.ordinal is
> before the first infinite.ordinal ω
> In particular, no infinite.ordinal is ω-1
>
> ----
> Consider ordinals i j k such that
> i∪{i} = j  and  j∪{j} = k
>
> Obviously, their order is  i < j < k
>
> Either they're all finite
> |i| < |j| < |k|
> or they're all infinite
> |i| = |j| = |k|
>
> No finite.to.infinite step exists.
> no visibleᵂᴹ finite.to.infinite step,
> no darkᵂᴹ finite.to.infinite step.
>
> Defining declares the meaning of one's words.
> 'Defining into existence' that which doesn't exist
> makes nonsense of whatever meaning one's words have.
>
> ⎛ if
> ⎜ g: j∪{j}→i∪{i}: 1.to.1
> ⎜ then
> ⎜ f(x) := (g(x)=i ? g(j) : g(x))
> ⎜ (Perl ternary conditional operator)
> ⎜ f: j→i: 1.to.1
> ⎜
> ⎜ if
> ⎜ f: j→i: 1.to.1
> ⎜ then
> ⎜ g(x) := (x=j ? i : f(x))
> ⎝ g: j∪{j}→i∪{i}: 1.to.1
>
> Therefore,
> i has fewer than j  iff  j has fewer than k
>
>>> ℕ has fewer elements than ℕ
>>
>> ℕ has ω-1 elements.
>
> ℕⁿᵒᵗᐧᵂᴹ holds all finite ordinals.
>
> Finite doesn't need to be small.
> ℕⁿᵒᵗᐧᵂᴹ holds ordinals which
> are big compared to Avogadroᴬᵛᵒᵍᵃᵈʳᵒ,
> but those big ordinals have an immediate predecessor,
> and each non.0.ordinal before them has
> an immediate predecessor.
> That makes them finite, but not necessarily small.
>
>>> Because ℕ does not have fewer elements than ℕ
>>> ℕ does not have fewer elements than ℕ∪{ℕ}
>>> and the rule of subsets is broken.
>>
>> ℕ = {1, 2, 3, ..., ω-1} = {1, 2, 3, ..., |ℕ|}
>
> ∀j ∈ ℕⁿᵒᵗᐧᵂᴹ:
> ∃k ∈ ℕⁿᵒᵗᐧᵂᴹ\{0}:
> k = j+1 ∧ ¬∃kₓ≠k: kₓ=j+1
>
> '+1': ℕⁿᵒᵗᐧᵂᴹ→ℕⁿᵒᵗᐧᵂᴹ\{0}: 1.to.1
> and the rule of subset is broken.
>
>

That's, ..., nice and all, yet, are you,
"preaching to the choir", or,
"reaching to the higher", the higher ground.

I.e., here it's not saying much.

Where's the "extra"-ordinary.

It's a matter of deductive inference there is one,
while the naive nicely arrives at it directly.