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

On 11/29/2024 4:29 PM, WM wrote:

On 29.11.2024 21:54, Jim Burns wrote:

On 11/29/2024 2:37 PM, WM wrote:

On 29.11.2024 19:08, Jim Burns wrote:

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JB contradicting himself:
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After all the swaps
 (of which no swap is a change in cardinality)
what remains is a proper subset
 (which is not a change in cardinality).

I (JB) contradict the finiteness of
the set of all finite cardinals.
Nothing more than that.
Change a set having a finite cardinality by 1.
The new set has a changed cardinality.
(This is essentially what 'finite cardinal' means).
The set of finite cardinalities,
for any finite cardinal,
includes more than that cardinal.
Each finite cardinal is not the one that set has.
Change the set of finite cardinalities by 1.
The new set has NOT changed cardinality.
The sets do not have any of
the cardinalities which would change.
The sets have a different cardinality,
  one which does not change when the set changes.
After all the swaps
  (of which no swap is a change in cardinality)
what remains is a proper subset
  (which is not a change in cardinality).
Because infinite.