Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 2024-12-16 11:28:52 +0000, WM said:

On 16.12.2024 11:23, Mikko wrote:

On 2024-12-15 11:33:15 +0000, WM said:

 

 We cannot name dark numbers as individuals.

 We needn't. The axioms of natural numbers ensure that every natural number
has a successor,

 The set, i.e. all numbers together, has no successor.

The set is not a natural number so can't be used as an argument
to the successor function. In the first order theory no set can
be used asn an argument to any function.
In set theory it is possible to define the successor function as
successor(X) = X ∪ {X}. That function has the required properties
if restricted to the sets that Cantor used for the representation
of the natural numbers but is applicable to every set.
--
Mikko