Re: The set of necessary FISONs

On 11.03.2025 18:11, Jim Burns wrote:

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.

Don't philosophize.

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)

Irrelevant.

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'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.

It is vague enough to admit your waffle. Therefore I use his
Z₀ = 0, {0}, {{0}}, ... denoted as sequence of numbers 0, 1, 2, 3, ....

Perhaps it displeases you that
Zermelo's isn't vague enough
to have room for your darkᵂᴹ numbers.

There is plenty of space for dark numbers after Z₀:
Take Cantor's ℕ. Delete 1. If you have deleted n, delete n+1. In all steps ℵ₀ numbers remain.

When we get see what we're intended to understand,
we get to see that Z₀ᵂᴹ doesn't hold darkᵂᴹ numbers.

In fact, the dark numbers are not in Z₀ but follow after Z₀.

 

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

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