Re: Replacement of Cardinality (ubiquitous ordinals)

On 8/2/2024 3:55 PM, Ross Finlayson wrote:

On 08/02/2024 03:39 AM, FromTheRafters wrote:

[...]

>
What I ask,
if that you surpass,
the inductive impasse,
of the infinite super-task.

I don't see what infinite super.task or
what inductive impasse you are asking about.
I see transfinite induction being
a consequence of ordinals being
well.ordered.
I see cisfinite induction being
a consequence of finite ordinals being
well.ordered and bounding only 2.ended sets.
----
In transfinite induction,
{ξ ∈ 𝕆: ¬P(ξ)} holds
no first ordinal β:
¬P(β) ∧ ¬∃α<β: ¬P(α)
⎛ Because ordinals are well.ordered,
⎝ the set is no.first only.if {ξ ∈ 𝕆: ¬P(ξ)} = {}.
¬∃β ∈ 𝕆:( ¬P(β) ∧ ¬∃α<β: ¬P(α) )
  ⟹
¬∃ξ ∈ 𝕆: ¬P(ξ)
∀β ∈ 𝕆:( P(β) ⇐ ∀α<β: P(α) )
  ⟹
∀ξ ∈ 𝕆: P(ξ)
----
⎛ Because a finite.ordinal bounds only
⎜ 2.ended non.{}.sets
⎜ finite.ordinal β bounds only
⎜ non.{} [0,α)ᴼ with a second.end
⎝ [0,α)ᴼ = [0,α-1]ᴼ  for 0 < α ≤ finite β
In cisfinite induction,
{ξ ∈ 𝕆ᶠⁱⁿ: ¬P(ξ)} doesn't hold
0 or ordinal β+1: ¬P(β+1) ∧ P(β)
...which implies
{ξ ∈ 𝕆ᶠⁱⁿ: ¬P(γ)} holds
no first ordinal β:
¬P(β) ∧ ¬∃α<β: ¬P(α)
and
no.first implies
{ξ ∈ 𝕆ᶠⁱⁿ: ¬P(ξ)} = {}.
P(0) ∧ ¬∃β ∈ 𝕆ᶠⁱⁿ: ¬P(β+1) ∧ P(β)
  ⟹
¬∃β ∈ 𝕆ᶠⁱⁿ:( ¬P(β) ∧ ¬∃α<β: ¬P(α) )
  ⟹
¬∃ξ ∈ 𝕆: ¬P(ξ)
P(0) ∧ ∀β ∈ 𝕆ᶠⁱⁿ: P(β) ⇒ P(β+1)
  ⟹
∀β ∈ 𝕆ᶠⁱⁿ:( P(β) ⇐ ∀α<β: P(α) )
  ⟹
∀ξ ∈ 𝕆ᶠⁱⁿ: P(ξ)
----

Then what *is* restricted comprehension?

>
Usually it's just
the antonym of expansion of comprehension.

I am more familiar with unrestricted comprehension
being the antonym of restricted comprehension.
Unrestricted comprehension grants that
{x:P(x)} exists because
description P(x) of its elements exists.
Restricted comprehension grants that
{x∈A:P(x)} exists because
description P(x) and set A exist.
The existence of set A might have been granted
because of Restricted.Comprehension or Infinity or
Power.Set or Union or Replacement or Pairing,
but A would be logically prior to {x∈A:P(x)}
by some route.