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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: More complex numbers than reals?
Date: Wed, 10 Jul 2024 17:16:00 -0700
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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:
>>>>>
>>>>>> 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?
>>
>> natural numbers: 1, 2, 3, ...
>>
>> 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...
>>>>
>>>> Fair enough? Or still crap? ;^o
>>>
>>> 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".
>>
>> 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 ...?
>