Re: More complex numbers than reals?

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

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

On 7/9/2024 4:45 PM, Ben Bacarisse wrote:

"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:
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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.
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    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.

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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 }
Both have the same number of elements, both have a monotonically increasing value wrt its elements wrt, ect...
Say a + 1 = b, b + 1 = c, c + 1 = d
We know that 3 + 1 = 4, 4 + 1 = 5 and 5 + 1 = 6
So, {a, b, c} and {3, 4, 5} share some interesting things, in a strange sense, so to speak.
{ a + 1 = b, b + 1 = c, c + 1 = d, ... }
{ 3 + 1 = 4, 4 + 1 = 5, 5 + 1 = 6, ... }
I see some relevant similarities between them. They are both infinite for sure. :^)
A side note:
WM would think that d and 6 are dark as in {a, b, c} does not show d, and { 3, 4, 5 } does not show 6.
Humm... ;^o

 

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

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So you are using WM's definition based on subsets?  That's a shame.  WM
is not a reasonable person to agree with!
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".

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The set of evens and odds has an infinite number of elements. Just like the
set of naturals.

 This sentence has nothing to do with what I wrote.  The set of evens and
the set {1,3,5} have no elements in common.  Both "miss out" all of the
elements of the other.  Which has "more" elements and why?  Can you
generalise to come up with a rule of |X| > |Y| if and only if ...?