Re: The set of necessary FISONs

On 3/8/2025 9:09 AM, WM wrote:

On 08.03.2025 12:58, Jim Burns wrote:

On 3/8/2025 3:45 AM, WM wrote:

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.

>

Zermelo's Axiom of Infinity describes Z
Z ∋ {} ∧ ∀a: Z ∋ a ⇒ Z ∋ {a}
Multiple sets satisfy that unique description.
That's an indefinite description.

>
My description is definite.

Your description is only definite because
ℕ ∋ n is definite.
What ℕ is,
the interesting part of your description of Z₀,
must be found elsewhere.

If only
{{{...{{{ }}}...}}} with finitely.many curly brackets
each with immediate.predecessor ⋃{{{...{{{ }}}...}}}
and with {} being one of its priors
are in Z₀
then, yes, that is a definite description.
>
And, yes, that seems to be what you meant.
If you ever want to make your descriptions clearer,
you won't be getting complaints from me.

>
That is exactly what I meant.
And that's also the induction of my argument
UF = ℕ  ==> Ø = ℕ.

A proof by induction
(a proof by this.inductive.subset.is.the.whole.set)
is only reliable for
a set which is its.own.only.inductive subset.
ℕ is its.own.only.inductive subset.
Z₀ is its.own.only.inductive subset.
⋃{F} union of the set {F} of all FISONs,
  is its.own.only.inductive subset.
For any set W which
is its.own.only.inductive subset (ℕ,Z₀,⋃{F},...),
∀j∈W:∃k∈W: j < k
Also, and equivalently,
¬∃k∈W:∀j∈W: j ≤ k
There is no last in W its.own.only.inductive.subset.
Your (WM's) argument is
⎛ (Matheologians say)
⎜ For each FISON F′ in {F},
⎜ the union ⋃{F:F′⊊F} of FISONs.after is unchanged
⎜ ∀F′∈{F}: ⋃{F:F′⊊F} = ⋃{F}
⎜
⎜ For the lastᵂᴹ (darkᵂᴹ) FISON F(ω-1),
⎜ the set {F:F(ω-1)⊊F} of FISONs.after is empty.
⎜ {F:F(ω-1)⊊F} = {}
⎜
⎜ Therefore,
⎝ ⋃{} = ⋃{F:F(ω-1)⊊F} = ⋃{F}
However,
{F} is its.own.only.inductive.subset.
No FISON in {F} is F(ω-1)
¬∃F″∈{F}:∀F′∈{F}: F′ ⊊ F″

If n ranges over only finite ordinals,
then
you didn't explicitly say that,

 Induction ranges over finite ordinals.

The set of {#A:#A<#Aᣕᵇ} of finite ordinals
is its.own.only.inductive.subset.
Induction is reliable for {#A:#A<#Aᣕᵇ}
but the rest of your argument fades away.

which explicitness is the reason for writing,
and,
if you had explicitly said that,
you should have said somewhere
what a finite ordinal is,

>
That is not under discussion here.

That has been under discussion for decades.
I think that these decades of discussion
have been, in large part, you assigning
different meanings to 'finite', etc.
and matheologians (among whom I place myself)
trying to discern what your meanings are.
Here's my best guess:
definableᵂᴹ  ==  finiteⁿᵒᵗᐧᵂᴹ  ==  #A<#Aᣕᵇ
darkᵂᴹ  ==  finiteⁿᵒᵗᐧᵂᴹ  ==  big and #A<#Aᣕᵇ
matheologicalᵂᴹ  ==  infiniteⁿᵒᵗᐧᵂᴹ  ==  #A=#Aᣕᵇ

n is usually denoting
a natural number.

Do we mean the same by 'natural number'?

FISONs are finite by definition.

Do we mean the same by 'finite'?