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

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.

 

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.

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. Try to express a natural number having less than ℵo sussessors. Impossible. Since the set of natural numbers cannot be subdivided into two actually infinite consecutive sets, the set of natural numbers that can be expressed is always finite whereupon almost all natural numbers are following.
Regards, WM