Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: More complex numbers than reals?
Date: Thu, 11 Jul 2024 17:14:05 +0200
Organization: A noiseless patient Spider
Lines: 58
Message-ID: <v6osru$2gom9$1@dont-email.me>
References: <v6ihi1$18sp0$6@dont-email.me> <87msmqrbaq.fsf@bsb.me.uk>
 <0dUETcjzkRZSIY0ZGKDH2IRJuYQ@jntp> <87v81epj5v.fsf@bsb.me.uk>
 <v6k216$1g6tr$3@dont-email.me> <878qyap1tg.fsf@bsb.me.uk>
 <v6mu4b$22opo$2@dont-email.me> <871q40olca.fsf@bsb.me.uk>
 <v6n880$23rgt$2@dont-email.me> <v6n8nt$2436h$3@dont-email.me>
 <v6n8v0$23rhi$3@dont-email.me> <v6n921$23rhi$4@dont-email.me>
Reply-To: invalid@example.invalid
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 7bit
Injection-Date: Thu, 11 Jul 2024 17:14:06 +0200 (CEST)
Injection-Info: dont-email.me; posting-host="2b43c9dd34f25362f42493cf5322921d";
	logging-data="2646729"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX1+fQmn72obGLjDL5KdRJh3h"
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:dFG7aiX3E81E2eswNBL+Fc3EI2c=
In-Reply-To: <v6n921$23rhi$4@dont-email.me>
Content-Language: de-DE
Bytes: 2959

Am 11.07.2024 um 02:29 schrieb Chris M. Thomasson:
> On 7/10/2024 5:28 PM, Chris M. Thomasson wrote:
>> On 7/10/2024 5:24 PM, Moebius wrote:

>>> If a = b = c, {a, b, c} still has "the same number of elements" as 
>>> {3, 4, 5 }? :-P

Hint: In this case card({a, b, c}) = 1.

Or with other words: Ex(x e {a, b, c} & Ay(y e {a, b, c} -> x = y)).

Using a special quantifier: E!x(x e {a, b, c}) ("There is exactly one x 
such that x is in {a, b, c}.")

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

You see (!) there terms in "{a, b, c}" (namely "a", "b" and "c") and 3 
terms in "{3, 4, 5}" (namely "3", "4" and "5").

> Then I start to examine how the elements are different and their 
> potential similarities, if any.

Right. In this case (i.e. an arithmetic context) we may safely assume 
that 3 =/= 3, 3 =/= 5 and 4 =/= 5. :-P

On the other hand, since we don't know anything concerning a, b and c, 
all we can state/say is:

1 <= card({a, b, c}) <= 3

(while card({3, 4, 5}) = 3.)

> For some reason, { a, b, c } and { 3, 4, 5 } makes me think of monotonically increasing.

Concerning { 3, 4, 5 } the reason is, that indeed 3 < 4 < 5, though { 3, 
4, 5 }  = { 5, 4, 3 } = ... etc.

But concerning { a, b, c } there simply is NO (rational) reason for 
assuming that.

a may be pi
b may be 0
c may be -i

Then {a, b, c} = {pi, 0, -i}. See?!

Or:

a may be 0
b may be 0
c may be 0

Then {a, b, c} = {0}. etc.

So how can we "compare" sets concerning their "size" (""number of 
elements"")? :-)

How about using a "measuring stick" (sort of)?