Re: The set of necessary FISONs

On 06.03.2025 20:03, Jim Burns wrote:

On 3/6/2025 12:05 PM, WM wrote:

On 06.03.2025 12:08, Jim Burns wrote:

On 3/6/2025 3:54 AM, WM wrote:

Am 06.03.2025 um 02:52 schrieb Jim Burns:

 

<<JB<WM>>>

>
You need not the intersection however because
Z₀ can also be defined by
{ } ∈ Z₀, and
if {{{...{{{ }}}...}}} with n curly brackets ∈ Z₀
then {{{...{{{ }}}...}}} with n+1 curly brackets ∈ Z₀.

>
No.

<</JB<WM>>>
>
No BECAUSE
that doesn't DEFINE Z₀

>
It defines Z₀ precisely.

 It does not exclude Bob.
 In order to exclude Bob,
some clause must state or imply Bob isn't in Z₀

No. The first element is the empty set. It does not contain Bob. All following elements are the empty set equipped with further curlybrackets.

 Induction says what's IN Z₀
More description is needed.

Therefore there is no Bob.

 

What does your inductive description permit in Z₀
which should not be permitted in Z₀ ?
Anything.
Your description permits Bob, as long as
{Bob}, {{Bob}}, {{{Bob}}}, ... are also in Z₀

>
Induction excludes Bob.

 I generalize 'end segment':

That doe not help. Induction does not talk about endsegments.

An unreliable argument is unacceptable.

Therefore I deleted your waffle.

Induction excludes Bob.

 Induction alone does not exclude Bob.

{ } ∈ Z₀, and if {{{...{{{ }}}...}}} with n curly brackets ∈ Z₀ then {{{...{{{ }}}...}}} with n+1 curly brackets ∈ Z₀. There is no slot for Bob!

Induction alone does not exclude Bob.

Waffle!
Find a difference between this induction and my induction about FISONs. Fail!
Regards, WM