Re: The set of necessary FISONs

On 11.02.2025 01:03, Richard Damon wrote:

On 2/10/25 10:00 AM, WM wrote:

Does Zermelo define a set by induction or only its elements?
>

Zermelo BUILDS a set by creating its members.

So do I.

Note, you keep on confusing a property of the members for a property of the set.

There is a set {1} smaller than ℕ.
If the set {1, 2, 3, ..., n} including its predecessors is smaller than ℕ, then the set {1, 2, 3, ..., n+1} including its predecessors is smaller than ℕ.

No individual FISON is required to build a set of FISONs whose union is the set of Natural Numbers,

No FISON is available. All FISONs belong to the inductive set F of insufficient FISONs. There is no FISON remaining.

but that doesn't mean the set of FISONs you CAN use is empty.

No, but it means: IF U(F) = ℕ, THEN { } = ℕ,
Regards, WM