Re: More complex numbers than reals?

Moebius <invalid@example.invalid> writes:

Am 11.07.2024 um 02:35 schrieb Ben Bacarisse:

Moebius <invalid@example.invalid> writes:
 

Am 11.07.2024 um 02:28 schrieb Chris M. Thomasson:

On 7/10/2024 5:24 PM, Moebius wrote:

Am 11.07.2024 um 02:16 schrieb Chris M. Thomasson:
>

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

>
HOW do you know that? Please define (for any sets A, B):
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      A and B /have the same number of elements/ iff ___________________ .
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(i.e. fill out the blanks). :-)
>
Hint: That's what Ben Bacarisse is asking for.
>
Sure, it's "obvious" for us. But how would you define "have the same
number of elements" (in mathematical terms) such that it can be DEDUCED
(!) für certain sets A and B?
>
________________________________________
>
Ok, I'm slighty vicious now... :-)
>
If a = b = c, {a, b, c} still has "the same number of elements" as {3,
4, 5 }? :-P

I see {a, b, c} and {3, 4, 5} and think three elements.

>
Even if a = b = c = 1?
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C'mon man! :-P

Please, that's a red herring, and you know it!  No where did I say that
a, b and c stood for anything (i.e. that they might be variables in the
maths sense).  I this sort of context they are just distinct symbols.

>
Of course, "a", "b" and "c" are three distinct symols. (Hell!)
>
But you dont't think that they should DENOTE some mathematical objects (in
a mathematical context)?*)

No, not without the context that says they are variables.

Huh?!
>
Are you doing math with symbols without any denotaton? Strange! (Really.)
[I'm sort of clueless now.]

Unless we allow symbols that just denote themselves, the only examples
we can conveniently give will be sets of numbers as, conventionally,
numerals are symbols that denote numbers.

Ok, let's reformulate my statement:
>
| Even if a = b = c?
>
Satisfied now?!

Not really.  We must be able write sets of things that are not numbers
without having to assume they are variables and therefore denote
something.  Otherwise we will always have to say a != b != c.  And in
this specific example we'd have to say even more since I wanted the two
sets to have no elements in common, so I'd have to have said a, b and c
don't stand for any of the numbers 1, 2 or 3.

I suppose since most people now know more about programming than maths,
I could have used strings:

  { "a", "b", "c" }

but to a mathematician that just looks clumsy and overly specific.

--
Ben.