Re: The set of necessary FISONs

On 3/4/2025 5:49 AM, WM wrote:

On 04.03.2025 11:42, Jim Burns wrote:

On 3/3/2025 4:56 PM, WM wrote:

On 03.03.2025 20:26, Jim Burns wrote:

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.

>
No.
Zermelo uses
induction which never goes beyond
a finite number of finite numbers.
Cantor claims
a number of finite numbers which is
larger than all finite numbers.

>
Yes.
Zermelo's ℕ and Cantor's ℕ are bracketed by
the union of FISONs and
the intersection of inductive subsets.
>
The union of FISONs and
the intersection of inductive subsets
are the same set.

----

⎜ During a typical Gish gallop,
⎜ the galloper confronts an opponent with a rapid series
⎜ of specious arguments,

>
Here is only *one* argument standing for a long while.

The union ⋃{F} of FISONs and
the intersection ⋂𝒫ⁱⁿᵈ of inductive subsets
are the same set.
Your (WM's) darkᵂᴹ numbers
are bracketed by ⋂𝒫ⁱⁿᵈ and ⋃{F}
They are in the empty difference ⋂𝒫ⁱⁿᵈ\⋃{F} = {}
They do not exist.
⎛ ⋂𝒫ⁱⁿᵈ = ⋃{F}
⎜
⎜⎛ ⋂𝒫ⁱⁿᵈ ⊆ ⋃{F}
⎜⎜
⎜⎜ All inductive sets have the same set ⋂𝒫ⁱⁿᵈ
⎜⎜ to be their intersection of inductive subsets.
⎜⎝ ⋃{F} is an inductive set.
⎜
⎜⎛ ⋃{F} ⊆ ⋂𝒫ⁱⁿᵈ
⎜⎜ k′ ∈ F′ ∈ {F}  ⇒  k′ ∈ ⋂𝒫ⁱⁿᵈ
⎜⎜
⎜⎜⎛ Otherwise,
⎜⎜⎜ if F′ ∋ kₓ ∉ ⋂𝒫ⁱⁿᵈ ∋ 0
⎜⎜⎜ then F′ ∋ (first) jₓ+1 ∉ ⋂𝒫ⁱⁿᵈ ∋ jₓ
⎜⎜⎜
⎜⎜⎜ if jₓ+1 ∉ ⋂𝒫ⁱⁿᵈ ∋ jₓ
⎜⎜⎜ then
⎜⎜⎜ ¬(jₓ ∈ ⋂𝒫ⁱⁿᵈ  ⇒  jₓ+1 ∈ ⋂𝒫ⁱⁿᵈ)
⎜⎜⎜ and ⋂𝒫ⁱⁿᵈ is not inductive.
⎜⎜⎜
⎜⎜⎝ However ⋂𝒫ⁱⁿᵈ is inductive.
⎜⎜
⎜⎜ k′ ∈ F′ ∈ {F}  ⇒  k′ ∈ ⋂𝒫ⁱⁿᵈ
⎜⎝ ⋃{F} ⊆ ⋂𝒫ⁱⁿᵈ
⎜
⎝ 𝒫ⁱⁿᵈ = ⋃{F}
----
https://en.wikipedia.org/wiki/Gish_gallop
⎛ Countering the Gish gallop
⎜
⎜ Mehdi Hasan, a British journalist, suggests using
⎜ three steps to beat the Gish gallop:
⎜ 1.
⎜ Because there are too many falsehoods to address,
⎜ it is wise to choose one as an example.
⎜ Choose the weakest, dumbest, most ludicrous argument
⎜ that the galloper has presented and
⎜ tear that argument to shreds ("the weak point rebuttal").
⎜ 2.
⎜ Do not budge from the issue or move on until
⎜ having decisively destroyed the nonsense and
⎜ clearly made the counter point.
⎜ 3.
⎜ Call out the strategy by name, saying:
⎜ "This is a strategy called the 'Gish Gallop'
⎜  —do not be fooled by the flood of nonsense
⎝ you have just heard."