Re: Incompleteness of Cantor's enumeration of the rational numbers

WM was thinking very hard :

On 18.11.2024 23:34, FromTheRafters wrote:

WM used his keyboard to write :

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

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If ℕ is an invariable set,

Sets don't change.

then it has an invariable number of elements. The "size" ℵo cannot express this.

That's likely why the terminology of size is IMO more appropriate when extended beyond the finite set's 'size' which is considered to be an actual count.

aleph_zero *IS* the cardinality of N.

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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.

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Only finite sets are countable.

Wrong, it is countable if there is a bijection. Some transfinite sets are countable.

But all "countably" infinite sets have the same size, which according to the different numbers of elements is a very unsharp measure.

That you don't like cardinal arithmetic, doesn't make cardinal arithmetic wrong.