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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Fri, 28 Feb 2025 18:04:33 +0100
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On 28.02.2025 16:52, Jim Burns wrote:
> On 2/28/2025 4:12 AM, WM wrote:
>> On 28.02.2025 01:00, Jim Burns wrote:
>>> On 2/27/2025 5:01 PM, WM wrote:
> 
>>> Zermelo's approach
>>> does not extend ∀n:Aᴺ(n) to Aᴺ(ℕ)
>>
>> It does.
> 
> Eppur si muove.
> 
>> Zermelo says it,
> 
> Nope.

Um aber die Existenz "unendlicher" Mengen zu sichern, bedürfen wir noch 
des folgenden, seinem wesentlichen Inhalte von Herrn Dedekind 
herrührenden Axioms

>> and it is easy to prove it:
>> Adding all natural numbers established the set ℕ.
> 
> We are finite beings. We do not do that.

Therefore we use FISONs without approaching ℕ.

>> How is Z accomplished?
> 
> Z is NOT accomplishedᵂᴹ.

The existence of Z is secured by induction: Um aber die Existenz 
"unendlicher" Mengen zu sichern, bedürfen wir noch des folgenden, seinem 
wesentlichen Inhalte von Herrn Dedekind herrührenden Axioms.

Without induction Z is not existing.
> 
> Zermelo describes the Z in the discussion

by induction. By what else?

>> { } and a ==> {a}.
> 
> True of Z because,
> when Zermelo describes Z,
> Zermelo describes such a set.

He describes it by induction.
> 
> Z being such a set is not induction.

The proof of existence is done by induction.
> 
> Induction proves inductive a subset of
> a set which is its.own.only.inductive.subset.

The set Z and all its inductive subsets are proven by induction.
> An inductive proof only proves about
> a set which is its.own.only.inductive.subset,
> like Z₀ and like ℕ, perhaps not like Z

Z contains many inductive subsets.
> 
>>> Proofs by induction are unreliable
>>> in Robinson arithmetic.
>>
>> Irrelevant.
> 
> What you (WM) think is a proof by induction
> is unreliable. But you don't care?

The set Z is not existing and not even defined without Zermelo's induction.

Note that for *definable* elements we have
{1, 2, 3, 4, 5} \ {1} \ {2} \ {3} \ {4} \ {5} = { },
which is same as
{1, 2, 3, 4, 5} \ {1, 2, 3, 4, 5} = { }.

Hence because of ∀n ∈ UA: |ℕ \ {1, 2, 3, ..., n}| = ℵo
we get
ℕ \ A(1) \ A(2) \ A(3) \ ... =/= { }
which is same as
ℕ \ UA =/= { }.

Regards, WM