Re: The set of necessary FISONs

Am Tue, 11 Feb 2025 11:05:16 +0100 schrieb WM:

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.

If by „predecessor” you mean {1, …, n-1} for n>1, then yes. N is not
finite and has no largest element.

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.

„Available”? Many FISONs exist. What is F? No single FISON suffices.

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

No

Then shut up.

but it means: IF U(F) = ℕ, THEN { } = ℕ,

U(F) != {}

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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.