Re: Simple enough for every reader?

On 2025-06-27 19:36:41 +0000, WM said:

On 27.06.2025 09:33, Mikko wrote:

On 2025-06-26 13:09:32 +0000, WM said:

 

 If we subtract in the order that is used for enumerating then a last one is necessary.

 No, there is no last one in an infinete enumeration.

 Then it is not finished or completed.

No, but it can be continued.

The word "infinite"
originally meant "having no end".

 That is true. But if infinite sets are complete, then they are infinite only because the end is not visible.

There is no mathematical meaning of "complete" that could be applied
to a set.

We can remove all odd numbers from the
natural numbers, leaving the even numbers, but there is no last number
removed.

 Then not all are removed. All completely, never ending and in order implies a contradiction.

All are removed when all are removed. That is not "never ending" as a
mathematical operation does not take any time.

For example, irrational numbers have their
arithmetic order but cannot be enumerated.

 The reason is that infinite sets cannot be enumerated. Also there completion would necessitate a last one.

 Infinite enumerable sets can. Then the enumeration is an infinte sequence.

 Then it is never completed. Going on and on is not a proof of completeness.

Being completed is not a mathematical concept. An infinite sequence just
is infinite.

< But they can be sutracted

from the set of real numbers and the result is the set of rational
numbers.

 

That can be done collectively only.

 Doesn't matter.

That is very important because these things are confused very often.

Confusions are best avoided by not using non-mathematical words and using
mathematical words only in their mathematical meanings.
--
Mikko