Re: Replacement of Cardinality

On 8/23/2024 10:28 AM, WM wrote:

Le 22/08/2024 à 16:22, FromTheRafters a écrit :

WM used his keyboard to write :

In a linear order of elements which
all have distances from each other,
there is necessarily a last one
(if nothing follows)
because the only alternative would be more than one.

>
You are thinking finitely again.

>
Logic is finite.

All finiteⁿᵒᵗᐧᵂᴹ ordered sets _and their subsets_
are 2.ended, except for {}.
An understanding of 'finiteⁿᵒᵗᐧᵂᴹ'
is essential to extending mathematicsⁿᵒᵗᐧᵂᴹ
from finite objects
to finite.length claims about infinite objects.
Claim Q₂ is not.first.false in
claim.sequence ⟨Q₁ Q₁⇒Q₂ Q₂⟩
⎛ Either Q₂ is not false,
⎜ or a claim before Q₂ (Q₁ or Q₁⇒Q₂) is false.
⎜ That there is no other possibility
⎝ can be demonstrated finitely.
Consider
the finite claim sequence ⟨Q₁ Q₂ ... Qₖ⟩ and
its sub.sequence ⟨F₁ F₂ ... Fⱼ⟩ of false claims
in ⟨Q₁ Q₂ ... Qₖ⟩
If we know ⟨F₁ F₂ ... Fⱼ⟩ is empty,
we know each claim in ⟨Q₁ Q₂ ... Qₖ⟩ is true.
⎛ Even if Qₖ is something like
⎜( Each natural ≥2 has a unique prime factorization
⎜ which no finite being can check individually,
⎝ we know that Qₖ is true.
If ⟨F₁ F₂ ... Fⱼ⟩ is non.empty,
we know ⟨F₁ F₂ ... Fⱼ⟩ is finiteⁿᵒᵗᐧᵂᴹ
and it has a first.end F₁
and F₁ is first.false in ⟨Q₁ Q₂ ... Qₖ⟩
⎛ We might not know which claim is first.false.
⎝ It's enough to know that a first.false exists.
If we know ⟨Q₁ Q₂ ... Qₖ⟩ is each not.first.false
we know its first.false F₁ doesn't exist
we know ⟨F₁ F₂ ... Fⱼ⟩ is empty
we know ⟨Q₁ Q₂ ... Qₖ⟩ is each true.
Even though Qₖ is about infinitely.many in ℕᵈᵉᶠ,
because
finite each.not.first.false ⟨Q₁ Q₂ ... Qₖ⟩ exists
we know that Qₖ is true of infinitely.many in ℕᵈᵉᶠ

Logic is finite.

Logic is a finite telescope with which
we can view (but not enter) the infinite realm.