Re: The set of necessary FISONs

On 14.02.2025 20:22, Jim Burns wrote:

On 2/14/2025 11:40 AM, WM wrote:

On 14.02.2025 16:42, Jim Burns wrote:

On 2/14/2025 7:52 AM, WM wrote:

 

Induction creates infinite sets.

>
Zermelo set theory [I...VII] describes
a domain of discourse with
Z an inductiveᶻ set

>
Where all elements are covered by induction.

 No,
we don't know that, in inductiveᶻ Z,
each element is covered by inductionᶻ.

Induction proves the existence of the inductive set of all its members.
Induction concerns the whole set. Compare Zermelo: "In order to secure the existence of infinite sets, we need the following axiom." [Zermelo: Untersuchungen über die Grundlagen der Mengenlehre I, S. 266] This is the axiom of infinity by induction: { } and with a also a'. It ascertains the existence of an infinite set. It ascertains the sets Z, Z_0 and the union of singletons ℕ.

Being covered by induction is
being in the only.inductive.subset.

That is here the set of all FISONs.

This reasoning only applies to
elements of the minimal.inductive set.
The minimal.inductive set is not
an element of the minimal.inductive set.
This reasoning does not apply to it.

The reasoning creates or describes this complete set.

Your alleged proof takes an unjustified leap
from "each FISON is omissible"
to "each set of omissible FISONs is ommissible".

No. Each FISON and all its predecessors are proven omissible by induction. This implies that the whole set is omissible.

 Each FISON is.
Some sets are and some sets aren't.

Which set do you have in mind?

 {F} is a set of omissible FISONs.
{F} isn't an omissible set of FISONs.
Without the leap, there is no conflict.

Without this leap there are no infinite sets. Remember "In order to secure the existence of infinite *sets*, we need the following axiom."
Regards, WM