Re: More complex numbers than reals?

On 7/10/2024 9:14 AM, WM wrote:

Le 10/07/2024 à 01:45, Ben Bacarisse a écrit :

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>

On 7/9/2024 10:30 AM, Ben Bacarisse wrote:

WM <wolfgang.mueckenheim@tha.de> writes:
>

Le 09/07/2024 à 14:37, Ben Bacarisse a écrit :
>

A mathematician, to whom this is a whole new topic, would start by
asking you what you mean by "more".  Without that, they could not
possibly answer you.

>
Good mathematicians could.
>

  So, what do you mean by "more" when applied to
sets like C and R?

>
Proper subsets have less elements than their supersets.

>
Let's see if Chris is using that definition.  I think he's cleverer than
you so he will probably want to be able to say that {1,2,3} has "more"
elements than {4,5}.

>
I was just thinking that there seems to be "more" reals than natural
numbers. Every natural number is a real, but not all reals are natural
numbers.

>
You are repeating yourself.  What do you mean by "more"?  Can you think
if a general rule -- a test maybe -- that could be applied to any two
set to find one which has more elements?

 No rule is better than a foolish rule, if it yields nonsense like Cantor's "bijections".

>

So, wrt the complex. Well... Every complex number has a x, or real
component. However, not every real has a y, or imaginary component...
>
Fair enough? Or still crap? ;^o

>
So you are using WM's definition based on subsets?  That's a shame.

 It is a reliable rule.
 

One consequence is that you can't say if the set of even numbers has
more or fewer elements than {1,3,5} because {1,3,5} is not a subset of
the even numbers, and the set of even numbers is not a subset of
{1,3,5}.  They just can't be compared using your (and WM's) notion of
"more".

 There are further rules. Every finite set has less elements than an infinite set.

What about an infinite number of finite sets? Say:
{ 1 }, { 1, 2 }, { 1, 2, 3 }, { 1, 2, 3, 4 }, ...

The set of rational numbers has 2|N|^2 + 1 elements. The set of real numbers is much larger than the set of rational numbers (but not because of Cantor's nonsense).
 Regards, WM