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

On 2024-11-29 13:53:39 +0000, WM said:

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.

What makes you think so?

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

 But it is reality. Without expressing your ideas you could not do mathematics.

But the mathematics would be the same anyway, independently of whether
anyone knows it.

If there is no concept, then it should be introduced.

The concept exsits but the thing does not.

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.

The finite set of expressed numbers has no role in mathematics, only the
infinite set of expressible numbers, and even that only in some contexts
like constructive mathematics.
--
Mikko