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From: Ben Bacarisse <ben@bsb.me.uk>
Newsgroups: sci.math
Subject: Re: More complex numbers than reals?
Date: Thu, 11 Jul 2024 00:53:25 +0100
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"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.

>>> 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 ...?

-- 
Ben.