Re: The set of necessary FISONs

On 3/3/2025 3:57 AM, WM wrote:

On 02.03.2025 21:28, Jim Burns wrote:

On 3/2/2025 1:52 PM, WM wrote:

On 02.03.2025 18:32, Jim Burns wrote:

On 3/2/2025 4:48 AM, WM wrote:

Induction abbreviates a supertask.
If 1 then 2, if 2 then 3, and so on.
But supertasks will never pass through
the dark numbers.

>
A claim about an indefinite element of Z₀ = ⋂𝒫ⁱⁿᵈ(Z)
cannot have a counter.example outside of Z₀
That is the source of a matheologian's certainty.

>
It has, namely ω, ω/2, etc.

>
ω is outside Z₀
ω cannot be a counter.example to
a claim about an indefinite element of Z₀

>
You are right.
I argued above concerning
Cantor's actually infinite ℕ.

Zermelo's ℕ and Cantor's ℕ are the same
up to isomorphism.
You refer to your (WM's) ℕ which holds
(darkᵂᴹ) elements which.cannot.be.said.to self.equal.
Elements like that are yours (WM's) alone.

Cantor's actually infinite ℕ.
It has
the undefinable elements ω/2, ω/10, ω/20 inside
and ω outside.

Zermelo's, Cantor's, and the matheologians' ℕ
is its.own.only.inductive.subset.
It does not change.
It does not end, not visiblyᵂᴹ, not darklyᵂᴹ.
It does not hold its least.upper.bound ω
It does not hold ω/2, ω/10, ω/20 which
(are alleged to) have finite multiples outside.

Z₀ does not contain undefinable elements.

Z₀ is its.own.only.inductive.subset.
Z₀ only holds elements with finite sets of priors.

Z₀ does not contain undefinable elements.
It contains simply all numbers that have FISONs
(like v. Neumann constructs the FISONs directly).

There are enough swaps within a set like Z₀,
which is its.own.only.inductive.subset  and
which only holds elements with finite sets of priors,
for Bob to not be anywhere after all the swaps.
No swap leaves Bob outside Z₀
No swap leaves Bob somewhere he doesn't leave later.
After all swaps,
Bob isn't anywhere he's left == anywhere.

ω/2 is outside Z₀ and outside the ordinals.
ω/2 cannot be a counter.example to
a claim about an indefinite element of Z₀, or to
a claim about an indefinite ordinal.

>
Right. All in Z₀ is definable.
∀n ∈ Z₀: |ℕ \ {1, 2, 3, ..., n}| = ℵo

∀n ∈ Z₀: |Z₀ \ {1, 2, 3, ..., n}| = ℵo