Re: The set of necessary FISONs

Am Sun, 16 Feb 2025 16:52:00 +0100 schrieb WM:

On 16.02.2025 12:59, joes wrote:

Am Sun, 16 Feb 2025 12:18:28 +0100 schrieb WM:

If all FISONs are omissible by induction, then the set of all FISONs
is omissible.

That is simply false.

Show what should remain.

What are you talking about? There are always inf. many if you remove
a natural number of FISONs, and their union is N.

And why? Note that you must define a first element.

Indeed, why should something remain if you remove everything?

Everything is removed that can be removed without changing the result.

You DO change the result to the empty set if you remove everything; the
union of all FISONs is not the empty set. You can only remove a finite
number of them, doesn’t matter which.

There is no doubt that this proves the whole set Z without assuming a
last element.

WDYM „prove a set”?

"In order to secure the existence of infinite sets, we need the
following axiom." [Zermelo] This is the axiom of infinity or induction:
{ } and if a then {a}.

You mean „prove the existence”.

 For every *element* you can prove the statement
„This, and only this segment can be omitted.” Unfortunately there is no
element that encompasses all numbers.

That element would have to be ω or N respectively.

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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.