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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Mon, 18 Nov 2024 17:34:49 -0500
Organization: Peripheral Visions
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WM used his keyboard to write :
> On 18.11.2024 22:53, FromTheRafters wrote:
>> on 11/18/2024, WM supposed :
>>> On 18.11.2024 18:11, FromTheRafters wrote:
>>>> WM pretended :
>>>>> On 17.11.2024 21:59, FromTheRafters wrote:
>>>>>
>>>>>> It is sometimes better to think of cardinality as a number indicating a 
>>>>>> notion of 'set size' rather than a notion of 'number many'. For finite 
>>>>>> sets these two notions merge.
>>>>>
>>>>> In the infinite they differ:
>>>>> ℵo is not the same as |ℕ| although |ℕ| is in the set of numbers 
>>>>> described by ℵo.
>>>>
>>>> aleph_zero is the same object as the cardinality of N.
>>>
>>> Yes, but |ℕ| is the number of elements of ℕ.
>> 
>> Or, more generally, it is the "SIZE" of the set. Size equals the number of 
>> elements in a *FINITE* set.
>> 
>> You said aleph_zero minus aleph_zero was different to the cardinality of N
>
> |ℕ| is the number of elements, not the cardinality!

That is why you have such difficulty in understanding infinite sets.

N is the set, and |N| is the size of the set.

>> aleph_zero *IS* the cardinality of N.
>
> Of course, but it is also the cardinality of all other countable sets which 
> have more or fewer elements.

No, only countably infinite sets. You have that all finite sets are 
countable too and have fewer elements.