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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: Sun, 15 Dec 2024 13:03:29 +0200
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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?

-- 
Mikko