Path: news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail
From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Sun, 9 Mar 2025 10:55:21 +0100
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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:
>>> 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.

No, my description is definite because every n can be obtained by 
addition of 1's (or of curly brackets). It is not necessary to have 
another definition of ℕ_def before.
> 
> What ℕ is,
> the interesting part of your description of Z₀,
> must be found elsewhere.

You are wrong. ℕ_def is the potentially infinite sequence of sums
1
1+1
1+1+1
....

ℕ is its completion, a set larger than all FISONs.
> 
>>> 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.

No. A proof by induction produces a set. If applied as I did above, the 
produced set 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.

Right.
> 
> For any set W which
> is its.own.only.inductive subset (ℕ,Z₀,⋃{F},...),
> ∀j∈W:∃k∈W: j < k

Yes, that is called potential infinity. A k larger than all j is not 
produced by induction. That is only produced by appointment.

> {F} is its.own.only.inductive.subset.
> No FISON in {F} is F(ω-1)

Right. ω and ω-1 cannot be attained by induction.
Therefore UF = ℕ  ==> Ø = ℕ.
>>> 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

Of course. We know also what we mean by ω, but it has no FISON. 
Therefore it is not a visible or definable number.

> darkᵂᴹ  ==  finite  ==  big

Look into the new thread "The truncated harmonic series diverges." to 
see that dark numbers are necessary. Having only definable denominators, 
the harmonic series could not diverge.

>> 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.
> 
>> 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.

Regards, WM