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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Tue, 11 Mar 2025 17:59:28 +0100
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On 11.03.2025 12:00, joes wrote:
> Am Mon, 10 Mar 2025 16:20:58 +0100 schrieb WM:
>> On 10.03.2025 09:30, joes wrote:
>>> Am Sun, 09 Mar 2025 20:13:53 +0100 schrieb WM:
>>
>>>> I am interested in the difference that you see between
>>>> Z₀ defined by { } ∈ Z₀, and if {{{...{{{ }}}...}}} with n curly
>>>> brackets ∈ Z₀ then {{{...{{{ }}}...}}} with n+1 curly brackets ∈ Z₀
>>>> and
>>>> The set of FISONs failing to have the union ℕ defied by induction:
>>>> |ℕ \ {1}| = ℵo, and if |ℕ \ {1, 2, 3, ..., n}| = ℵo then |ℕ \ {1, 2,
>>>> 3, ..., n+1}| = ℵo.
>>>
>>> How you can view the first set not to have the same union (by Neumann's
>>> equivalence) as the the second one would be most welcome. In any case,
>>> the second set does not exist according to your contradictory specifi-
>>> cation. They are clearly isomorphic.
>>
>> That is no my opinion but Jim Burns' opinion.
>> Both sets are identical. Both are potentially infinite collections.

> No, they are not finite.

They are no finite.

> You can't believe Z_0 to be "complete" in
> your sense if you don't think the second set is (and accept that they
> are equivalent).

They are minutely equivalent. To described both take ℕ. Delete 1. If you 
have deleted n, delete n+1. In all steps ℵ₀ numbers remain.

Regards, WM