Re: The set of necessary FISONs

Am Sun, 02 Feb 2025 12:25:49 +0100 schrieb WM:

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.

It’s not an assumption. Hint: F(n+1) = F(n) u n+1 and then
U({F(n) for all n e N}).

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.

The union of a nonempty set is not empty.

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]

But P(ω) does not hold.

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

There are no naturals with infinite segments.

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.