Re: The set of necessary FISONs

On 3/11/2025 4:28 AM, WM wrote:

On 10.03.2025 23:07, Jim Burns wrote:

I thought you (WM) might be interested to hear that
we can prove that proofs.by.induction are reliable
for FISONs.

>
That is trivial.

Sadly,
you (WM) are wrong to think you know
what that sentence means.
Even if you sound as though you agree,
even if you write that sentence in smoke
across the skies of Augsburg,
even if you die with it on your lips,
what it means to you (WM)
is not
what I want to tell you.

I am interested in the difference you see between
>
Zermelo's Z₀ defined or ensurede

The difference between 'defined' and 'ensured'
is that
Zermelo defines Z (Z₀ is after) to be
an inductive set, and so Z is inductive,
but that definition doesn't ensure
that Z exists.
Separately from the defining of Z
Zermelo's Axiom of Infinity ensures
that something exists which satisfies
the definition of Z
The axiom doesn't create Z
Imagine all domain.candidates of
Zermelo's set theory
on a table in front of you.
Then Zermelo's Infinity is proclaimed,
and the candidates without Z
disappear from the table.
They are not candidates any longer.
You pick a domain off the table.
You know something you can call Z is in it.
The proclaiming of Infinity ensured
that the domain you picked holds
some Z.
The proclaiming of Infinity didn't create Z
Z was always in the domain you picked.
It didn't create the domain you picked.
That domain was always on the table.
If we stretch, we maybe could say that
the proclaiming of Infinity
UNcreated the domains without any Z
But, be careful! Those domains
only not.exist in the sense that
we aren't discussing them -- today.
On another day? We'll have to see.

Zermelo's Z₀ defined or ensurede by induction:
{ } ∈ Z₀,
and if
{{{...{{{ }}}...}}} with n curly brackets ∈ Z₀
then
{{{...{{{ }}}...}}} with n+1 curly brackets ∈ Z₀.

You (WM) are defining your own (not Zermelo's) Z₀ᵂᴹ
Perhaps you feel that Zermelo's definition is
more complicated than needed.
Perhaps it displeases you that
Zermelo's isn't vague enough
to have room for your darkᵂᴹ numbers.
We are intended to understand by
your use of the variable name 'n' that
⎛ Z₀ᵂᴹ is linearly ordered
⎜ {} is in Z₀ᵂᴹ and first
⎜ no element in Z₀ᵂᴹ is last.
⎜ for each split of Z₀ᵂᴹ with
⎜ foresplit and hindsplit both non.empty
⎜ its foresplit ends at i  and
⎜ its hindsplit begins at j,
⎝ i and j such that  {i} = j
When we get see what we're intended to understand,
we get to see that Z₀ᵂᴹ doesn't hold darkᵂᴹ numbers.

and
>
the the set F of removable FISONs
defined or ensured by induction.
ℕ \ F(1) = ℵo,
and if
ℕ \ F(1) \ F(2) \ F(3) \ ... \ F(n) = ℵo
then
ℕ \ F(1) \ F(2) \ F(3) \ ... \ F(n+1) = ℵo.

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