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

On 11/29/24 8:53 AM, WM wrote:

On 29.11.2024 11:12, Mikko wrote:

On 2024-11-28 12:31:35 +0000, WM said:

 

No. For every unit fraction there exist infinitely many smaller unit fractions, infinitely many of which cannot be expressed. They remain simply to be smaller.

>
The number 0 can be expressed with a finite string. The successor of an
expresssible number can be expressed. The ratio of two expressible numbers
can be expressed. Nothing else is a rational number.

>
That is potential infinity. Actual or completed infinity is a fixed quantity larger than every natural number. The set of expressed numbers however remains finite in every instance.

>
In mathematics everything is actual and nothing is potential.

 Then you could avoid the ℵo unit fractions beneath every unit fraction. But you cannot.
 

WHy would you need to? They exist, so are there, and are all definable.

There is no
concept of "expressed", only "expressible".

 But it is reality. Without expressing your ideas you could not do mathematics. If there is no concept, then it should be introduced.

It may be a physical fact if a number has actually been expressed, but expressing a number doesn't change it, so whether a number has been expressed or not is irrelevant.

 

You can't change anything
about numbers. There are infinitely many finitely expressible numbers and
that includes all rationals.

 All expressed numbers are a finite set. It can be increased without bound but it is never actually infinite like the remainder in ℕ.
According to Cantor actual infinity is a fixed quantity.

Yes, but the set of expressed numbers has no meaning to mathematics, as no relevent property of a number depends on it being expressed

Regrads, WM
 

Perhaps you could work out a concept of "Expressed Mathematics", but it will have the problem that its fundamental basis, the set of expressed numbers, is a variable, not a constant.
This is the sort of thing that lead to the downfall of naive set theory, so maybe all you are doing is showing that your naive mathematics, that is based on this concept of expressed numbers, is just broken.