Re: Replacement of Cardinality

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, ....