Path: ...!news.nobody.at!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: "Chris M. Thomasson" Newsgroups: sci.math Subject: Re: More complex numbers than reals? Date: Wed, 10 Jul 2024 14:23:23 -0700 Organization: A noiseless patient Spider Lines: 58 Message-ID: References: <87msmqrbaq.fsf@bsb.me.uk> <0dUETcjzkRZSIY0ZGKDH2IRJuYQ@jntp> <87v81epj5v.fsf@bsb.me.uk> <878qyap1tg.fsf@bsb.me.uk> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Wed, 10 Jul 2024 23:23:24 +0200 (CEST) Injection-Info: dont-email.me; posting-host="9a69ddc669a6ef7ff2bcd6fc06fda374"; logging-data="2188088"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19tEYtcbIMEOhVo7cJr6nlN7L4KWIgq+Go=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:49eGZBMjcxpF32PJP1/EojHgETA= In-Reply-To: <878qyap1tg.fsf@bsb.me.uk> Content-Language: en-US Bytes: 3405 On 7/9/2024 4:45 PM, Ben Bacarisse wrote: > "Chris M. Thomasson" writes: > >> On 7/9/2024 10:30 AM, Ben Bacarisse wrote: >>> WM 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... > >> 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.