Re: The set of necessary FISONs

On 3/5/2025 4:22 AM, WM wrote:

On 04.03.2025 23:05, Jim Burns wrote:

On 3/4/2025 2:56 PM, WM wrote:

On 04.03.2025 17:43, Jim Burns wrote:

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

Therefore ℕ is more than UF.
By induction like Zermelo we find:
∀n ∈ U(F): |ℕ \ {1, 2, 3, ..., n}| = ℵo.

>
∀n ∈ ⋃{F}: |⋃{F}\{1,2,3,...,n}| = ℵ₀

>
No.
⋃{F} is wrong because F is the set of FISONs.

⎛ I have been writing
⎜ F′ a FISON
⎜ F″ a possibly.different FISON
⎜ {F} the set of all FISONs
⎝ {F:F′⊆F} the set of F′ and all FISONs.after
⎛ I have been writing
⎜ ⋃{F:F′⊆F} = ⋃{F}
⎜ for
⎝ "The FISONs before F′ are omissible"

⋃{F} = F.

⎛ {F} = {{},{0},{0,1},{0,1,2},...}
⎝ ⋃{F} = {0,1,2,3,...}

∀n ∈ U(F): |ℕ \ {1, 2, 3, ..., n}| = ℵo.

>
∀n ∈ ⋃{F}: |⋃{F}\{1,2,3,...,n}| = ℵ₀

>
No.
⋃{F} is wrong because F is the set of FISONs.
UF does not contain
an n that is larger than all n by ℵ₀.

Writing U{F} for UF (?),
yes,
U{F} does not contain
an n that is larger than all n by ℵ₀.
Also:
∀n ∈ ⋃{F}: |⋃{F}\{1,2,3,...,n}| = ℵ₀
which is another way of saying
∀F′ ∈ {F}:
¬∃F″ ∈ {F}: #(⋃{F}\F′) < #F"

⎛ Assume ⋃{F} is finite.

>
UF is (potentially in-) finite.

∀ˢᵉᵗA:
  #A < ℵ₀  ⇔
  ∃F′ ∈ {F}: #A < #F′
¬∃F′ ∈ {F}: #⋃{F} < #F′
¬(#⋃{F} < ℵ₀)
#⋃{F} ≥ ℵ₀
Also,
∀F′ ∈ {F}:
¬∃F″ ∈ {F}: #(⋃{F}\F′) < #F"
∀F′ ∈ {F}:
¬(#(⋃{F}\F′) < ℵ₀)
#(⋃{F}\F′) ≥ ℵ₀

It is constructed by induction

Construction is proof of existence.
Induction is,
for a set which is
its.own.only.inductive.subset,
proof for a subset that it's inductive,
and thus is the only inductive subset,
which is the whole set.
Induction is only for
sets which are
their.own.only.inductive.subsets.

which is variable and
therefore
has no largest number
but never reaches a completion.

A construction is a proof of existence.
A proof is
a finite sequence of finite.length claims
(each claim true.or.not.first.false).
A (finite) proof completes.
A construction completes.