Re: The set of necessary FISONs

WM <invalid@no.org> wrote:

Am 19.02.2025 um 15:50 schrieb joes:

Am Wed, 19 Feb 2025 15:31:54 +0100 schrieb WM:

They didn’t say that by induction, you can infer properties of a set
from its elements - like being finite or not. Compare {Q, N, R},
the set of endsegments, and {0, 1, 2}.

By induction all elements can be defined. This guarantees the existence
of an infinite set. Um aber die Existenz "unendlicher" Mengen zu
sichern, bedürfen wir noch des folgenden ... Axioms. [Zermelo:
Untersuchungen über die Grundlagen der Mengenlehre I, S. 266]

„If every natural is finite, then there are only finitely many
naturals.”?

Word on this?

The set of all natural numbers which can be defined is (potentially in-)
finite. infinitesimally smaller than ℕ.

The set of all natural numbers is N, by definition.  N is infinite.
Lose the useless qulifications.  They're not helping anybody.

Proof: If UF = ℕ is assumed, then F(1) can be omitted without changing
the union of the remainder.

I can accept this, though it needs a bit more rigour.  Under what
conditions can a FISON be omitted from a union of them without changing
that union?  Only when there is a subsequent FISON in the union.

And if F(n) can be omitted without changing this union, then also
F(n+1) can be omitted without changing this union.

This is questionable,indeed.  What precisely is the nature of the
relationship between F(n) and F(n+1) which allows this?  There would
appear to be none.  You have failed to prove this inductive step.
Again, you can omit F(n+1) when and only when there is a bigger FISON in
the union.

That makes the omitted FISONs the inductive collection of all FISONs
and proves the implication: If UF = ℕ, then { } = ℕ..

You have entirely failed to prove that.  A problem is that you are
defining members of a set only by their relationship to other members.
On finishing your alleged induction, that relationship no longer holds -
You end up with a set of FISONs none of which can be omitted, since
there isn't a bigger FISON in the complementary set of remainders.  So
the whole mechanism collapses like a bubble bursting.

I think you started this thread wanting to prove a whimsical
contradiction here which you would then use to establish the falsehood
of something or other.  Then you got bogged down trying to insist the
falsehood was true.

A large part of the problem is that you are using FISONs as though they
were some sort of primitive.  See how long ago it is since you actually
defined what you mean by FISON on this group, if you ever have.  Using
FISONs as primitives is almost guaranteed to engender confusion, which
seems to be why you use them.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).