Re: The set of necessary FISONs

Am 18.02.2025 um 18:14 schrieb Jim Burns:

For inductive sets with multiple.inductive.subsets,
it's inadequate to prove that a subset is inductive
in order to conclude that subset is the whole set.

The FISONs are an inductive set with only one inductive subset.

The subset of {F} with that property
is inductive.
Showing that a subset is inductive
is called proof by induction.

That is what I do.

 {F} is the only.inductive.subset of {F}.
In the case of {F}, a proof by induction shows
that any subset of {F} with that property is {F},
because that subset can't be anything else..

That reasoning is silent about
whether the _set_ (not its elements) has A(k).

The set without any element is empty.
The elements are defined by induction in order to guarantee the 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]
Regards, WM
Regards, WM