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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Mon, 3 Feb 2025 13:41:41 +0100
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On 02.02.2025 19:23, Jim Burns wrote:

>> F(1) can be discarded.
>> If F(n) can be discarded, then F(n+1) can be discarded.
>>
>> Note:
>> Mathematical induction is a method for proving that
>> a statement P(n) is true for every natural number n
>> that is, that
>> the infinitely many cases P(0),P(1),P(2),P(3),...
>> all hold. [Wikipedia]
> 
> For each k ∈ ⋃{FISON}:

No, for *all* k ∈ ⋃{FISON}.

Peano creates the set ℕ by induction.
I remove the set of FISONs by the same induction:
n ==> n+1.

What us the difference?
> ⋃{FISON} = ⋃({FISON}\{F(k)})
> 
> P(k)  :⇔  ⋃{FISON}=⋃({FISON}\{F(k)})
> 
> For each k ∈ ⋃{FISON}:  P(k)
> 
> Because
> none in linearly.ordered {FISON} is last.

None of the natural numbers is last.
How can Peano create the complete set by induction?

Regards, WM