Re: The set of necessary FISONs

On 3/9/2025 5:55 AM, WM wrote:

On 08.03.2025 20:27, Jim Burns wrote:

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

On 08.03.2025 12:58, Jim Burns wrote:

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  ==  finite

Compare what you quote to what I wrote:
definableᵂᴹ  ==  finiteⁿᵒᵗᐧᵂᴹ  ==  #A<#Aᣕᵇ

[...]
>

darkᵂᴹ  ==  finite  ==  big

Compare what you quote to what I wrote:
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'?

>
There are two different meanings:
All positive integers having FISONs or all positive integers.

The distinction which you (WM)
have been dodging for decades
is between
sets which have
fuller.by.one and emptier.by.one counterparts
_which are a different size_ (finiteⁿᵒᵗᐧᵂᴹ)
and
sets which have
fuller.by.one and emptier.by.one counterparts
_which are the same size_ (infiniteⁿᵒᵗᐧᵂᴹ)

FISONs are finite by definition.

>
Do we mean the same by 'finite'?

>
A natural number n is finite.
It is an integer between 0 and ω: 0 < n < ω.
Sometimes 0 is included, never ω is included.

You erased the distinction I made and
you haven't given any replacement.
(Your ωᵂᴹ is merely a big finite.)
#A size of set A
#Aᣕᵇ size of fuller.by.one set, Aᣕᵇ = A∪{b} ≠ A
#A < #Aᣕᵇ larger fuller.by.one set: A finite
#Y = #Yᣕᶻ same.size fuller.by.one set: Y infinite
The distinction infinite.v.finite is useful
because
not all sets are finite,
and the distinction, for fuller.by.one sets,
of same.v.different sizes leads to
very different properties.
A notable example of an infinite (ie, #Y = #Yᣕᶻ) set
is the set of finite set.sizes {#A:#A<#Aᣕᵇ}
⎛ For each finite.set.size ξ in {#A:#A<#Aᣕᵇ},
⎜ there is a larger subset #{#A<ξ+1:#A<#Aᣕᵇ} = ξ+1
⎜
⎜ {#A:#A<#Aᣕᵇ} doesn't contain
⎜ a subset larger than {#A:#A<#Aᣕᵇ}
⎜
⎜ For each finite.set.size ξ in {#A:#A<#Aᣕᵇ},
⎝ ξ isn't the size of {#A:#A<#Aᣕᵇ}
{#A:#A<#Aᣕᵇ} is what we mean by ℕ,
what you mean by ℕ_def.
----

My description is definite.

>
Your description is only definite because
ℕ ∋ n is definite.

>
No, my description is definite
because
every n can be obtained by addition of 1's
(or of curly brackets).

Finitely.many 1's or curly.brackets.
How many is that?
Your decades.long argument has been about
how many that is --
but you avoid saying what you mean.
You even avoid hearing what we mean.