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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
Date: Mon, 16 Dec 2024 12:23:46 +0200
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On 2024-12-15 11:33:15 +0000, WM said:

> On 15.12.2024 12:03, Mikko wrote:
>> On 2024-12-14 09:50:52 +0000, WM said:
>> 
>>> On 14.12.2024 09:52, Mikko wrote:
>>>> On 2024-12-12 22:06:58 +0000, WM said:
>>> 
>>>>>>> In mathematics, a set A is Dedekind-infinite (named after the German 
>>>>>>> mathematician Richard Dedekind) if some proper subset B of A is 
>>>>>>> equinumerous to A. [Wikipedia].
>>>>>> 
>>>>>> Do you happen to know any set that is Dedekind-infinite?
>>>>>> 
>>>>> No, there is no such set.
>>>> 
>>>> The set of natural numbers, if there is any such set,
>>> 
>>> If ℕ is a set, i.e. if it is complete such that all numbers can be used 
>>> for indexing sequences or in other mappings, then it can also be 
>>> exhausted such that no element remains. Then the sequence of 
>>> intersections of endsegments
>>> E(1), E(1)∩E(2), E(1)∩E(2)∩E(3), ...
>>> loses all content. Then, by the law
>>> ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
>>> the content must become finite.
>>> 
>>>> is Dedekind-infinte:
>>>> the successor function is a bijection between the set of all natural
>>>> numbers and non-zero natural numbers.
>>> 
>>> This "bijection" appears possible but it is not.
>> 
>> So you say that there is a natural number that does not have a next
>> natural number. What number is that?
> 
> We cannot name dark numbers as individuals.

We needn't. The axioms of natural numbers ensure that every natural number
has a successor, no natural number is its own successor, and no two natural
numbers has the same successor. If that is not possible then there are no
natural numbers.

-- 
Mikko