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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Sun, 2 Feb 2025 12:25:49 +0100
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On 01.02.2025 20:21, Jim Burns wrote:
> On 2/1/2025 7:56 AM, WM wrote:

>> There is the assumption that
>> a set with U(F(n)) = ℕ exists.
>> Without changing the union
>> we can remove every element by induction.
>> No element remains.
>> The set does not exist.
> 
> Each finiteᵒᵘʳ initial segment F(k) of ⋃{F(n)}
> can grow¹ to another initial segment F(k+1)
> which is also finiteᵒᵘʳ, and is larger than F(k),
> and is not larger than ⋃{F(n)}
> 
> {F(n}} holds each finiteᵒᵘʳ initial segment F(k)
> ⋃{F(n)} is larger than each F(k).

But all F(n) can be discarded without changing the union.
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]

Therefore if U(F(n)) = ℕ, then  { } = ℕ

Regards, WM