Re: The set of necessary FISONs

On 3/11/2025 2:01 PM, WM wrote:

On 11.03.2025 18:11, Jim Burns wrote:

to be
an inductive set, and so Z is inductive,
but that definition doesn't ensure
that Z exists.

>
He defines Z by induction in order to ensure
the existence of an infinite set.

Zermelo defines Z to be an inductive set.
Zermelo asserts the Axiom of Infinity which
restricts candidates for the domain of discussion
to those which contain an inductive set.
Zermelo does both, but
they're not the same thing.
Proof.by.induction proves,
from a subset ⊆ Y being inductive,
that that subset = the whole set Y.
A proof.by.induction is only reliable in a set Y in which
an inductive subset can only be the whole subset.
A set such as Y = ⋂𝒫ⁱⁿᵈ(Z), for Z inductive, for example.
Defining Z₀ to be inductive is insufficient
to prove, from a subset ⊆ Z₀ being inductive,
that the subset = the whole set Z₀

|A| < ℵ₀  ∧  |B| = ℵ₀  ⇒  |B\A| = |B| = ℵ₀

>
|Z₀| < ℵ₀

No.
ℵ₀ is the size of the set of finite sizes.
ℵ₀ = |{|A|:|A|<|Aᣕᵇ|}|
Z₀ is inductive.
Z₀ ⊇ ⋂𝒫ⁱⁿᵈ(Z₀) its.own.only.inductive.subset
|⋂𝒫ⁱⁿᵈ(Z₀)| = ℵ₀
ℵ₀ = |⋂𝒫ⁱⁿᵈ(Z₀)| ≤ |Z₀|
⎛ |⋂𝒫ⁱⁿᵈ(Z₀)| = ℵ₀
⎜ because
⎜
⎜ ⋂𝒫ⁱⁿᵈ(Z₀) is its.own.only.inductive.subset.
⎜ {|A|:|A|<|Aᣕᵇ|} is its.own.only.inductive.subset.
⎜
⎜ For each two.ended initial segment of ⋂𝒫ⁱⁿᵈ(Z₀)
⎜ there is a two.ended initial segment of {|A|:|A|<|Aᣕᵇ|}
⎜ which is larger.
⎜
⎜ {|A|:|A|<|Aᣕᵇ|} is not smaller than ⋂𝒫ⁱⁿᵈ(Z₀)
⎜
⎜ For each two.ended initial segment of {|A|:|A|<|Aᣕᵇ|}
⎜ there is a two.ended initial segment of ⋂𝒫ⁱⁿᵈ(Z₀)
⎜ which is larger.
⎜
⎜ ⋂𝒫ⁱⁿᵈ(Z₀) is not smaller than {|A|:|A|<|Aᣕᵇ|}
⎜
⎝ |⋂𝒫ⁱⁿᵈ(Z₀)| = |{|A|:|A|<|Aᣕᵇ|}| = ℵ₀

|Z₀| < ℵ₀  ∧  |ℕ| = ℵ₀  ⇒  |ℕ\Z₀| = |ℕ| = ℵ₀
>
|UF| < ℵ₀  ∧  |ℕ| = ℵ₀  ⇒  |ℕ\UF| = |ℕ| = ℵ₀
>
UF = ℕ  ==> Ø = ℕ