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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
Date: Tue, 07 Jan 2025 15:01:16 -0500
Organization: Peripheral Visions
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WM formulated the question :
> On 07.01.2025 14:49, FromTheRafters wrote:
>> WM was thinking very hard :
>>> On 07.01.2025 02:36, Ross Finlayson wrote:
>>>> On 01/06/2025 02:43 PM, Jim Burns wrote:
>>>
>>>>> It would be great if you (WM) did NOT
>>>>> find lemma 1 weird,
>>>>> but it is what it is.
>>>
>>> It is not weird. But your conclusions are weird.
>>>
>>>> The inductive set being covered by
>>>> initial segments is an _axiom_ of ZF.
>>>
>>> And the existence of the set ℕ is also an axiom of ZF.
>> 
>> Which axiom?
>
> The axiom of infinite set. The elements of the infinite set have been 
> designed after Dedekind as Zermelo, the creator of the axiom himself 
> confesses. Its elements can be called cardinal numbers.

No, that set is a superset and the naturals need to be extracted from 
it.