Re: The set of necessary FISONs

On 3/10/2025 4:42 AM, WM wrote:

On 10.03.2025 00:11, Jim Burns wrote:

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

<JB>

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

</JB>

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

>
Where?

For example, here:
⎛ A  FISON is linearly ordered,
⎜ begins at 0, ends at a FISON.end, and,
⎜ for each split,
⎜ its foresplit ends at i or is empty  and
⎜ its hindsplit begins at j or is empty,
⎝ i and j such that i+1 = j
[1]
That describes infinitely.many natural numbers
without describing any _particular_ natural number,
like 1 or 2 or 3 in {0,1,2,3,...}
What we can do with [1] which
we can't do with {0,1,2,3,...}
is
follow it with not.first.false claims,
which we can see are each not.first.false,
which we can see without understanding them,
the way we don't understand Q in ⟨ P P⇒Q Q ⟩
In a finite sequence of claims in which
each claim is true.or.not.first.false,
each claim is true.
We see and know they're true
_even if we don't understand the claims_
But it starts with a description,
the more detailed, the more useful.
I haven't re.stated [1] recently,
maybe not even in this year,
but I have stated and re.stated
[1] and variations on that theme,
over and over,
past the point where you (WM) complain.
That description isn't the usual description.
I'd guess the Peano axioms are the usual.
But you (WM) dispute the Peano axioms,
so I cooked up a description of
the same things Peano describes which
I hoped would align more with
your (WM's) sense of what.we.are.discussing.
Re.re.re.statements aren't the usual practice.
Often a description "somewhere else" is in
a journal or a lecture from decades earlier,
and the reader, if they want them,
gets to do the finding of them themselves.

For the first,
you aren't using
any definition which fills
a role comparable to "FISON'.

>
Wrong.
{{ }} = {1},
{{{ }}} = {1, 2},
{{{{ }}}} = {1, 2, 3}.

You've defined 3 numbers and stopped.
Follow that with
not.first.false claims about 1,2,3.
They will be true claims
...about 1,2,3.
Logic don't care.
But you might.
That's not very impressive, is it?
Peano and Zermelo and Cantor and I
describe infinitely.many
and follow it with claims about infinitely.many
which we don't need to understand in order to
see that they are true.
You can also do that, but
it starts with some unspecific description
somewhere.

I can see that you aren't because
you think you have defined 'finite' there,

>
All FISONs are finite.

Look at [1].
I told you
what it means to be finite.
You need to refer to the same being done
somehow somewhere,
do it yourself now,
or settle for discussing only 1,2,3.

somehow (by a darkᵂᴹ definition?),
and because
you have said explicitly that you don't need
a definition elsewhere.

>
Before you recognized that
you have run out of counterarguments
you never doubted that definition.

I have never used {0,1,2,3,...}
as a definition.
It might have taken me a while to realize
that you think {0,1,2,3,...} is a definition,
because
none of this works like that.

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.

>
I use definable natumbers
as everybody knows how to use them.

Non.specific description then not.first.false
unlocks infinity.