Re: Replacement of Cardinality (ubiquitous ordinals)

On 08/03/2024 12:08 PM, Jim Burns wrote:

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.
>
>

Geometry, axiomatic geometry or Euclid's,
is a classical theory, and it's constructive,
there's only expansion of comprehension,
while it's let out that line-drawing or continuity
itself is undefined, instead the elements are given,
then they all make any constructions possible,
in expansion of comprehension.
This is among reasons why some people say that
well-founded set theory, is non-constructive, and
that various of its results are not constructivist.
Now, I'm a structuralist and a constructivist in
a strong mathematical platonism, while you have
declared that nominalist fictionalism is more your
cup of tea, then that ZF's usual axioms that restrict
comprehension, Well-Foundedness as a regularity
and Well-Founded Infinity as an escape, are your
implicit statements before any other kind of otherwise
primitive statement, as in "see rule number one".
Yet, "formalism" isn't necessarily the either camp,
that though formalism is structure.
Often the Well-Foundedness is called Regularity,
it's rulial, an exception of sorts, while, there are
other kinds of Regularity like Well-Ordering, then
as with regards to whether Well-Ordering is restriction
of comprehension, has that the Well-Founded Ordinary
Infinity so given: already always has an ordinal assignment
so there's no disputing it, in ZF, ZFC.
Formalism, mathematical conscientiousness:  24/7/365.
Then here I built line-reals as a usual sort of continuum
limit then make a neat only-diagonal argument that as
a function it's non-Cartesian, with about the same approach
as what establishes the anti-diagonal the un-countability,
this is the "both", in ZF, and ZFC.
You can add it to rule one.  Which is getting sort of longer, ....
Then in a theory of ubiquitous ordinals instead,
then ZF/ZFC is just models of fixed-points in fragments/extensions.
So, classical geometry is a classical and constructivist theory, ....
Zeno has theories, not paradoxes.