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: Tue, 17 Nov 2020 17:25:30 +0000 Organization: A noiseless patient Spider Lines: 31 Message-ID: <87pn4bj3p1.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="a3f68d594c737541b9396ae63b92e4a6"; logging-data="14607"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX18dOXUL9Lyj8Xu9gVfurbnv+tIM74ZpQbM=" Cancel-Lock: sha1:GI4N078doCh3ZyBp4PlIURgyhVA= sha1:WhMQ67r3J/YiCwZEpEDOZOR/DGA= X-BSB-Auth: 1.48bced948cfe1d3ab8a1.20201117172530GMT.87pn4bj3p1.fsf@bsb.me.uk Xref: csiph.com comp.programming:11468 Richard Heathfield writes: > 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. That makes it sound like it is the infinitude itself that leads to the conclusion, but there are lots of infinite sets with a largest element. > 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. This is what leads to the conclusion: the axioms for the > relation. > Therefore I is not > the biggest integer, even if we (quite improperly) assign ∞ to I. (Oh, and hi. Haven't seen you around for ages.) -- Ben.