Path: csiph.com!eternal-september.org!feeder.eternal-september.org!reader02.eternal-september.org!.POSTED!not-for-mail From: Ben Bacarisse Newsgroups: comp.programming Subject: Re: Theorem: There are at least =?utf-8?Q?=E2=88=9E^4?= integers in 2 dimensions. Date: Wed, 18 Nov 2020 15:44:19 +0000 Organization: A noiseless patient Spider Lines: 55 Message-ID: <87blfuoejw.fsf@bsb.me.uk> References: Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: 8bit Injection-Info: reader02.eternal-september.org; posting-host="00e4dfa80ab5f3594322e8267b52436d"; logging-data="5741"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1/yqU69MgWKqg3DGVnald1Qw5PVWT9F11Q=" Cancel-Lock: sha1:DA1ZqW+AvKIZfwX4aU+rUNw7pkQ= sha1:6haT+eCAG3univCUw6Xanhm+Kbg= X-BSB-Auth: 1.ab5ad238ed0ae8a0f83b.20201118154419GMT.87blfuoejw.fsf@bsb.me.uk Xref: csiph.com comp.programming:11480 David Brown writes: > On 18/11/2020 13:50, Richard Heathfield wrote: >> On 17/11/2020 21:36, David Brown wrote: >>> On 17/11/2020 14:10, Richard Heathfield wrote: >>>> On 17/11/2020 06:09, K R wrote: >>>>> Lets look shapes with 3,4,5,6,... angles. >>>>> They have inside ∞ crosses. Because by dividing areas with 2, we get ∞ >>>>> points. >>>>> Lets begin with 2 line connected shapes with 3 and 4 angles. >>>>> We can continue to N shapes. >>>>> Now lets start to go around these shapes with connecting shapes. >>>>> Process can continue forever. >>>>> So we get number N*∞*∞*∞ points. >>>>> The biggest integer is ∞, so there are ∞^4 integers. >>>> >>>> No, ∞ is not an integer. There are infinitely many integers, and >>>> therefore there can be no biggest integer. This is easily proved by >>>> reductio. If we assume there is a biggest integer I, we can immediately >>>> disprove the assumption by adding 1 to I to form a new integer J that is >>>> 1 larger than I. Therefore I is not the biggest integer, even if we >>>> (quite improperly) assign ∞ to I. >>>> >>> >>> It's nice to see someone trying to post something other than spam in >>> this group (and nice to hear you are still around, Richard). >>> Unfortunately, you are responding to a spambot, not a person, and the >>> original post made no sense. >>> >>> If you'd like to talk about infinities, however, I'm game - it's always >>> a fun subject (though only vaguely on-topic here).  We could start by >>> proving that ∞⁴ is equal to ∞. >> >> I just tried to do that, but what I actually proved (in text that I've >> now deleted) is that there are as many fourth powers of natural numbers >> as there are natural numbers, which is (a) obvious, and (b) not quite >> the same thing as proving that ∞⁴ = ∞ - or at least, if it *is* the same >> thing, I don't quite see why. >> > > Well, if we want to prove this kind of thing, it's important to be > precise. And that means dumping ∞, because that symbol is used to mean > various things (such as increasing integers or reals without a limit, or > when defining limits, or to mean "infinity" without saying /which/ > infinity). And let's start with ∞² rather than ∞⁴. > > The cardinality (size) of the set of natural numbers ℕ = {0, 1, 2, …} is > ℵ₀. (I hope the unicode comes through okay when you read it - that's > aleph 0.) It is the smallest transfinite cardinality. ∞ is ambiguous. It could mean ℵ₀ or it could mean 𝜔, the smallest infinite ordinal. 𝜔⁴ is just 𝜔𝜔𝜔𝜔 ≠ 𝜔. -- Ben.