Re: More complex numbers than reals?

After serious thinking Moebius wrote :

Am 11.07.2024 um 12:34 schrieb FromTheRafters:

Ben Bacarisse was thinking very hard :

Moebius <invalid@example.invalid> writes:
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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:
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{a, b, c} vs { 3, 4, 5 }
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Both have the same number of elements, [...]

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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). :-)
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Hint: That's what Ben Bacarisse is asking for.
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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?
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________________________________________
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Ok, I'm slighty vicious now... :-)
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If a = b = c, {a, b, c} still has "the same number of elements" as {3,
4, 5 }? :-P

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

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

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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.

 Indeed!

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Nonsense. (See below.)

You have stated this nonsense before, I simply disagree with a set in roster form having duplicates.

I sometimes try to steer WM away from 'math' symbols in sets like asking for a bijection of something like {elephant, rhinoceros, dune buggy} and {circle, square, megaphone}.

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Here you used well estabished /names/ (constants) for certain objects which - as is well known - are not identical. With other words,
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elephant =/= rhinoceros
elephant =/= dune buggy
rhinoceros =/= dune buggy,
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etc.
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Just using DIFFERENT (but arbitrary) symbols, say "a", "b", "c", does not ensure for that (i.e. a =/= b, a =/= c, b =/= c).
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Actually, even a = b = c is POSSIBLE in this case. (Leading to card({a, b, c}) = 1.)
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Now consider the two (different) names ("symbols") "turtle", "chelonian".
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Does {turtle, chelonian} contain 2 elements, huh?!

Roster form is also known as enumeration notation and one should be able to, with small finite sets, simply count the elements to obtain cardinality.
Turtle and chelonian are two different words representing that which might be equivalent in some sense, so yes cardinality two.