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

On 2024-11-30 12:30:20 +0000, WM said:

On 30.11.2024 10:54, Mikko wrote:

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

 

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?

 The fact that every unit fraction that can be defined has ℵo unit fractions beneath it proves that you cannot avoid that fact.

Which fact? And why that fact would be relevant to Cantor's enumeration?

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.

 Anyone uses it. It is deplorable that most don't know what they need.

Whether someone or anyone uses it or not, the mathematics is the same.

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,

 That is always finite as proved by fact.

Doesn't matter as it has that has no mathematical consequences.
--
Mikko