Re: The set of necessary FISONs

On 3/9/2025 3:13 PM, WM wrote:

On 09.03.2025 17:26, Jim Burns wrote:

On 3/8/2025 2:27 PM, Jim Burns wrote:

Here's my best guess:
definableᵂᴹ  ==  finiteⁿᵒᵗᐧᵂᴹ  ==  #A<#Aᣕᵇ
darkᵂᴹ  ==  finiteⁿᵒᵗᐧᵂᴹ  ==  big and #A<#Aᣕᵇ
matheologicalᵂᴹ  ==  infiniteⁿᵒᵗᐧᵂᴹ  ==  #A=#Aᣕᵇ

You even avoid hearing what we mean.

>
I am interested in
the difference
that you see between
>
Z₀ defined by { } ∈ Z₀, and
if {{{...{{{ }}}...}}} with n curly brackets ∈ Z₀
then {{{...{{{ }}}...}}} with n+1 curly brackets ∈ Z₀
>
and
>
The set of FISONs failing to have
the union ℕ defied by induction:
|ℕ \ {1}| = ℵo, and
if |ℕ \ {1, 2, 3, ..., n}| = ℵo
then |ℕ \ {1, 2, 3, ..., n+1}| = ℵo.

For the second,
I have elsewhere provided you with
the description of a FISON
which is needed to make sense of
what is NOT
⎛ a proof of inductivity for a subset of
⎜ a set which we already knew is
⎝ its.own.only.inductive.subset.
For the first,
you aren't using
any definition which fills
a role comparable to "FISON'.
I can see that you aren't because
you think you have defined 'finite' there,
somehow (by a darkᵂᴹ definition?),
and because
you have said explicitly that you don't need
a definition elsewhere.
You (WM) think you don't need to say
what natural number is,
even where you clearly have taken the term
and used it in your own, private way.