Re: The set of necessary FISONs

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

On 03.03.2025 20:26, Jim Burns wrote:

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.
⎛ For W inductive, Y inductive,
⎜ W ⊇ W∩Y inductive
⎜ ⋂𝒫ⁱⁿᵈ(W∩Y) = ⋂𝒫ⁱⁿᵈ(W)
⎜
⎜ Similarly,
⎜ Y ⊇ W∩Y inductive
⎜ ⋂𝒫ⁱⁿᵈ(W∩Y) = ⋂𝒫ⁱⁿᵈ(Y)
⎜
⎜ ⋂𝒫ⁱⁿᵈ(Y) = ⋂𝒫ⁱⁿᵈ(W∩Y) = ⋂𝒫ⁱⁿᵈ(W) = ⋂𝒫ⁱⁿᵈ
⎜
⎝ V inductive  ⇒  ⋂𝒫ⁱⁿᵈ(V) = ⋂𝒫ⁱⁿᵈ ⊆ V
⎛ union ⋃{F} of FISONs  is inductive
⎜
⎜ ⋃{F} inductive  ⇒  ⋂𝒫ⁱⁿᵈ(⋃{F}) = ⋂𝒫ⁱⁿᵈ ⊆ ⋃{F}
⎜
⎝ ⋂𝒫ⁱⁿᵈ ⊆ ⋃{F}
⎛ For each FISON F′ ∈ {F}
⎜ for each k′ ∈ F′:  k′ ∈ ⋂𝒫ⁱⁿᵈ
⎜
⎜⎛ Assume otherwise
⎜⎜ Assume kₓ ∈ F′ ∈ {F} and kₓ ∉ ⋂𝒫ⁱⁿᵈ
⎜⎜
⎜⎜ FISON F′ is well.ordered
⎜⎜ exists first jₓ+1 ∈ F′ such that
⎜⎜ jₓ+1 ∉ ⋂𝒫ⁱⁿᵈ and jₓ ∈ ⋂𝒫ⁱⁿᵈ
⎜⎜
⎜⎜ However,
⎜⎜ ⋂𝒫ⁱⁿᵈ is inductive
⎜⎜ jₓ ∈ ⋂𝒫ⁱⁿᵈ  ⇒  jₓ+1 ∈ ⋂𝒫ⁱⁿᵈ
⎜⎜ ¬(jₓ ∈ ⋂𝒫ⁱⁿᵈ  ∧  jₓ+1 ∉ ⋂𝒫ⁱⁿᵈ)
⎜⎝ Contradiction.
⎜
⎜ For each FISON F′ ∈ {F}
⎜ for each k′ ∈ F′:  k′ ∈ ⋂𝒫ⁱⁿᵈ
⎜
⎜ For each FISON F′: F′ ⊆ ⋂𝒫ⁱⁿᵈ
⎜
⎝ ⋃{F} ⊆ ⋂𝒫ⁱⁿᵈ
⎛ ⋂𝒫ⁱⁿᵈ ⊆ ⋃{F}
⎜
⎜ ⋃{F} ⊆ ⋂𝒫ⁱⁿᵈ
⎜
⎝ ⋂𝒫ⁱⁿᵈ = ⋃{F}

But that is not the topic.
The topic is this:
In exactly the same way as Z₀ is constructed by its elements,
the set of removable FISONs
is constructed by its elements,
namely by induction.

https://en.wikipedia.org/wiki/Gish_gallop
⎛ Gish gallop
⎜
⎜ [...]
⎜ During a typical Gish gallop,
⎜ the galloper confronts an opponent with a rapid series
⎜ of specious arguments, half-truths, misrepresentations
⎜ and outright lies, making it impossible for the opponent
⎜ to refute all of them within the format of the debate.
⎜ Each point raised by the Gish galloper
⎜ takes considerably longer to refute than to assert.
⎜
⎜ [...]
⎜
⎜ The difference in effort between making claims and
⎜ refuting them is known as Brandolini's law or
⎜ informally "the bullshit asymmetry principle".
⎝ Another example is firehose of falsehoods.