Re: More complex numbers than reals?

On 7/10/2024 5:46 PM, Ben Bacarisse wrote:

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

On 7/10/2024 4:53 PM, Ben Bacarisse wrote:

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
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On 7/9/2024 4:45 PM, Ben Bacarisse wrote:

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
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On 7/9/2024 10:30 AM, Ben Bacarisse wrote:

WM <wolfgang.mueckenheim@tha.de> writes:
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Le 09/07/2024 à 14:37, Ben Bacarisse a écrit :
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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.

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Good mathematicians could.
>

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

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Proper subsets have less elements than their supersets.

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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?

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natural numbers: 1, 2, 3, ...
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Well, it missed an infinite number of reals between 1 and 2. So, the reals
are denser than the naturals. Fair enough? It just seems to have "more", so
to speak. Perhaps using the word "more" is just wrong. However, the density
of an infinity makes sense to me. Not sure why, it just does...

I am trying to get you to come up with a definition.  If it is all about
"missing" things then you can't compare the sizes of sets like {a,b,c}
and {3,4,5} as both "miss" all of the members of the others.

>
{a, b, c} vs { 3, 4, 5 }
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Both have the same number of elements,

 That will fall down for infinite sets unless, by decree, you state that
your meaning of "more" makes all infinite sets have the same number of
elements. 

What about dropping the word more in favor of density, or granularity of an infinite set? The reals are denser, or more granular than the rationals and reals? Crap?

You can do that if you want -- the definition is yours -- but
it does not match your initial suspicions.  For example, you probably
think, intuitively, that there are "more" reals then integers.
 

both have a monotonically increasing
value wrt its elements wrt, ect...

 Now that's an interesting start, more interesting than you probably know
right now.  You are tying "more" to the idea of an ordering.  But what
happens when there is no obvious first element?  For example are there
more reals in (0,1) than in (1,2)?  What about (0,1) and (1,3)?