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NNTP-Posting-Date: Sun, 28 Jul 2024 23:32:40 +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 16:32:58 -0700
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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.