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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Thu, 6 Feb 2025 20:32:53 +0100
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On 06.02.2025 19:54, Jim Burns wrote:
> On 2/6/2025 11:55 AM, WM wrote:
>> On 06.02.2025 15:57, Jim Burns wrote:
> 
>>> The key is that  ∀ᴺ¹n: ∃ᴺ¹j′: n<j′
>>
>> The key is that
>> the set ℕ is created by induction.
> 
> The set ℕ₁ is described as having induction valid for it.

Then it is the collection ℕ_def of definable numbers.
> 
> Sets missing natural numbers and
> sets with extra, non.inducible, un.natural numbers
> are not ℕ₁

Then it is the set ℕ of all natural numbers.

You contradict yourself.
> 
>> If the set M is described as the smallest set satisfying
>> 1 ∈ M and n ∈ M ==> n+1 ∈ M
>> then ℕ\M = Ø.
> 
> ℕ₁ = ∅ satisfies that definition.

No. 1 is not in ∅,
> 
> Better:
> ℕ₁ is the emptiest set M such that
> 1 ∈ M and n ∈ M ⇒ n+1 ∈ M
> Thus:
> 1 ∈ ℕ₁ and n ∈ ℕ₁ ⇒ n+1 ∈ ℕ₁
> ∀P:(1 ∈ P and n ∈ P ⇒ n+1 ∈ P) ⇒ ℕ₁ ⊆ P
> 
> Is ℕ₁ the emptiest set M such that
> 1 ∈ M and n ∈ M ⇒ n+1 ∈ M ?

Relevant is the set of FISONs.
>> I prefer Wikipedia:
>> ∀P( P(1) /\ ∀k(P(k) ==> P(k+1)) ==> ∀n (P(n)).
> 
> That's intended to be part of the definition of ℕ₁

As well it is the definition of the collection of all FISONs.
> 
> Which is curious, when one considers that
> ℕ₁ appears nowhere in it.

The axiom of induction holds for all predicates P which satisfy induction.

If the set M is described as the smallest set satisfying
F(1) ∈ M and F(n) ∈ M ==> F(n+1) ∈ M
then M contains all FISONs which can be subtracted from U(Fn)) without 
changing the assumed result ℕ.

Regards, WM