Re: how

On 6/22/2024 4:12 PM, WM wrote:

Le 22/06/2024 à 19:34, Jim Burns a écrit :

On 6/22/2024 8:11 AM, WM wrote:

But every FISON is a very, very proper subset:
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
That statement covers all FISONs.

>
U{FISON} = ⋂{inductive}
>
∀n ∈ U{FISON}: |U{FISON}\{1,2,3,...,n}| = ℵ₀

 Each FISON is Union of all predecessors.
U{FISON} is a FISON.

Yes,
U{FISON} is the union of all its predecessors.
No,
U{FISON} is not a FISON.
Each ordinal, finite or transfinite, is
the union of all its predecessors.
Inaccessible κ is
the union of all its predecessors.
https://en.wikipedia.org/wiki/Inaccessible_cardinal
U{FISON} is the union of all its predecessors.
U{FISON} is not a FISON. It is not finite.
Nonzero _finite_ ordinal k
  has an immediate predecessor k⁻¹  and
each nonzero predecessor j<k
  has an immediate predecessor i=j⁻¹  and
0 is a predecessor.
0≤k  ∧  ∀j: 0<j≤k ⇒ ∃i: 0≤i<k ∧ i⁺¹=j

No FISON has ℵ₀ elements.

U{FISON} is not a FISON. It is not finite.

No union of FISONs has ℵ₀ elements.

ℵ₀ is defined to be how many elements U{FISON} has.
Your claim is cousin to the claim that
no triangle has three corners.