Re: The set of necessary FISONs

On 2/22/2025 5:17 AM, WM wrote:

On 21.02.2025 20:40, Jim Burns wrote:

However,
all sets Z which satisfy Infinityᶻᶠᶜ
HAVE A SUBSET which
is usable in a proof.by.inductionᶻᶠᶜ

>
Zermelo calls that set Z_0 or set of numbers.

The usable SUBSET ℕⁱᵒᵒⁱˢˢ of Zermelo's asserted Z
is its.own.only.inductive.sub.set (iooiss).
Our sets do not change.
I'm pretty sure your ℕ_def = ℕⁱᵒᵒⁱˢˢ
Our sets do not change.
Because ℕⁱᵒᵒⁱˢˢ is iooiss,
proving a subset is inductive  is
proving that  that inductive.subset is
its.own.only.inductive.sub.set,
which is ℕⁱᵒᵒⁱˢˢ
Our sets do not change.
What does 'inductive' mean?
There are multiple answers.
Each 'inductive' has its own ℕⁱᵒᵒⁱˢˢ
about which a proof.by.induction proves.
⎛ You (WM) say 'inductive S' means
⎜ ∀k:S∋k⇒S∋k+1 ∧ S∋1
⎜ Then
⎜ ℕ₁ⁱᵒᵒⁱˢˢ = {1,2,3,...}
⎜
⎜ We (matheologists) say 'inductive S' means
⎜ ∀k:S∋k⇒S∋k+1 ∧ S∋0
⎜ Then
⎜ ℕ₀ⁱᵒᵒⁱˢˢ = {0,1,2,...}
⎜
⎜ If 'inductive S' means
⎜ ∀k:S∋k⇒S∋k+1 ∧ S∋137
⎜ Then
⎜ ℕ₁₃₇ⁱᵒᵒⁱˢˢ = {137,138,139,...}
⎜
⎜ If 'inductive S' means
⎜ S∋1 ∧ ∀k:S∋k⇒S∋2⋅k
⎜ Then
⎜ Powers(2)ⁱᵒᵒⁱˢˢ = {1,2,4,8,16,...}
⎜
⎜ If 'inductive S' means
⎜ S∋2 ∧ ∀k:S∋k⇒S∋min.{i∈[k,k!+1]ᴺ:[2,k]ᵉᵃᶜʰ̷|i}
⎜ Then
⎝ Primesⁱᵒᵒⁱˢˢ = {2,3,5,7,11,...}
For each two meanings 'inductiveₓ' and 'inductiveᵥ',
there is a bijection f between ℕₓⁱᵒᵒⁱˢˢ and ℕᵥⁱᵒᵒⁱˢˢ
⎛ firstₓ  ᶠ↦  firstᵥ
⎜ if ℕₓ ∋ x  ᶠ↦  v ∈ ℕᵥ
⎝ then sucₓ(x)  ᶠ↦  sucᵥ(v)
The map f is one.to.one because
we only consider versions of 'inductive'
such that sucₓ(⋅) and sucᵥ(⋅) are one.to.one.
The map is onto ℕᵥ because
the image f(ℕₓ) ⊆ ℕᵥ is inductive.
f(ℕₓ) can only be ℕᵥⁱᵒᵒⁱˢˢ
One.to.one, onto, bijective.
For each two meanings 'inductiveₓ' and 'inductiveᵥ',
there is a bijection f between ℕₓⁱᵒᵒⁱˢˢ and ℕᵥⁱᵒᵒⁱˢˢ
which is what
  ℕₓⁱᵒᵒⁱˢˢ and ℕᵥⁱᵒᵒⁱˢˢ are _the same size_
means to us.
You (WM) can refuse to say "the same size".
It makes what you're saying less clear,
but less clarity is probably your point.
The bijection isn't sent away
by refusing to say "same size".

We can remove all numbers from Z_0 and produce the empty set.

We can remove all numbers
from {137} and produce the empty set.
{137} ≠ 137

Homework: Show the same for the set of FISONs.