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NNTP-Posting-Date: Sun, 28 Jul 2024 23:42:02 +0000
Subject: Re: Replacement of Cardinality
Newsgroups: sci.logic,sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sun, 28 Jul 2024 16:42:10 -0700
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On 07/28/2024 04:32 PM, Ross Finlayson wrote:
> On 07/28/2024 04:25 PM, Ross Finlayson wrote:
>> 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.
>>
>>
>
> Foundations is more than a field.
>

Now, if there is something as relevant as Cardinality,
as primary, for mathematical foundations, it's: Continuity,
that Continuity, is so essentially primary, fundamental,
central, and ubiquitous, makes for the Cardinality as
next to Ordinality for counting vis-a-vis Numbering,
in where there are various (and perhaps, nowhere only
"standard") models of integers, where Cohen for the
Independence of the Continuum Hypothesis in Cardinals
makes an extra-ordinary bit of model there courtesy
a pretty simple induction about Ordinals vis-a-vis Cardinals
in a theory with numbering vis-a-vis counting that there
is: the extra-ordinary, about ubiquitous ordinals
in any old theory.

That there is one at all, ....