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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Tue, 11 Feb 2025 11:05:16 +0100
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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