Path: csiph.com!eternal-september.org!feeder.eternal-september.org!reader01.eternal-september.org!.POSTED!not-for-mail From: Keith Thompson Newsgroups: sci.logic,sci.math,comp.theory,comp.ai.philosophy Subject: Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Date: Tue, 17 Sep 2019 11:42:57 -0700 Organization: None to speak of Lines: 120 Message-ID: References: <87blvlequ8.fsf@bsb.me.uk> <4bmdnZCR0LAZOuLAnZ2dnUU7-RfNnZ2d@giganews.com>

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Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: 8bit Injection-Info: reader02.eternal-september.org; posting-host="8286bf823aa591a4447bcf77f162785a"; logging-data="8725"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19/EJn/t82/0APTWxPGVfKY" User-Agent: Gnus/5.13 (Gnus v5.13) Emacs/23.1 (gnu/linux) Cancel-Lock: sha1:n2vQCcZqsJV950r8KC6SZMyrrKw= sha1:vfUBrbwaO8QRfd749cfAs+jY0ZI= Xref: csiph.com sci.logic:210937 sci.math:458586 comp.theory:19952 comp.ai.philosophy:19258 Mike Terry writes: > On 17/09/2019 05:38, Keith Thompson wrote: >> André G. Isaak writes: >> [...] >>> Yeah, the account I offered in my earlier post was kind of mangled. My bad. >>> >>> The main point I was trying to get across with that post was that the >>> real numbers in a given interval can't be ordered as smallest, second >>> smallest, third smallest, etc. I shouldn't really have claimed that >>> there was no 'ordering' among reals - that was a rather unfortunate >>> choice of terms, but rather that there is nothing akin to a successor >>> relation in terms of which they can be ordered. >>> >>> It is possible for an interval to have a smallest element, but it is >>> certainly not guaranteed. In a closed interval, there will be a smallest >>> element but only because the interval is defined as including its >>> greatest lower bound. For an open interval, this is not the case. >>> >>> I wanted to get him away from trying to derive the open interval from >>> the closed interval and instead to focus on the defining properties of >>> the intervals simply because he is determined to impose something akin >>> to a successor relation on his intervals despite the fact that no such >>> relation exists. >>> >>> You're correct, though, that this probably isn't going to actually >>> clarify things for Pete. >> >> Thanks for the clarification. >> >> Here's my attempt to refute the idea that (A,B] has a least element. >> I'm not trying to argue with Pete (though I expect he'll see this and >> respond to it). >> >> Briefly, if (0,1] has a least member, what about the result of dividing >> that member by 2? >> >> In more detail, (0,1] is the half-open interval consisting of >> the real numbers between 0 and 1, including 1 but excluding 0. >> Assume that this interval has a least element, an element x such >> that for each member m of the interval, either x> >> Let y=x/2. If we're dealing with real numbers, then 0> is a member of the interval and is less than x -- contradicting >> the original assertion that x is the least element. Conclusion: >> the interval (a set of real numbers) has no least element. >> >> Arbitrarily inventing a new notation, let _(0,1]_ be a set of, um, >> something or other. It includes all the real numbers in (0,1], >> and it may also include some other things, each of which is greater >> than 0, less than or equal to 1, and somewhere on the "number line" >> between 0 and 1. Some of the non-real elements might be points, >> or infinitesimals. I'll assert that for any two members of *this* >> set, exactly one of ab is true. >> >> I can imagine that *this* set has a least element. For example >> we can define "iota" as an infinitesimal, something that is not a >> real number but that has a location on the number line, directly >> adjacent to the real number 0. It is greater than 0 and less than >> any positive real number. >> >> For the same argument I used to demonstrate that x is not the least >> member of the interval *not* to apply to iota, one of the following >> must be true: >> >> - iota/2 <= 0 >> - iota/2 >= iota >> - iota/2 is not defined >> >> Though we can define something like this, I'm having difficulty >> thinking of a way to define it that is consistent *and useful*. >> We need the non-real members of the interval to behave radically >> differently under common arithmetic operations. Not that there's >> anything wrong with that, but it seems to require inventing a set >> that's even denser than the reals, but is nevertheless in some >> vague sense discrete rather than continuous. As far as I can tell >> (and it's very likely I've missed something), most proposed systems >> of infinitesimals do not have this property. > > There are properly defined concepts such as hyper-real numbers, and > Conway's sureal numbers: > > > > > and in both cases, the equivalent of your iota is divisible by 2 in a > reasonable way, and so as you correctly reason, there is still no "next > point" following a given point. Also, it should be obvious that both of > these systems of numbers are uncountable, as they both contain a subset > isomorphic to the set of real numbers. It sounds to me (based on my extensive research consisting of a few minutes on Wikipedia) that PO has invented something similar to dual numbers , except that the coefficient of ε (epsilon) is an integer rather than a real number and certain operations like division are (arbitrarily?) stated to be undefined. Without directly speculating on PO's motivation, I'll note that I see no use for this other than to make it possible for an open interval over some superset of the reals to have a least member, a more defensible variant of PO's original incorrect claim that an open interval over the reals has a least member. > IMO, just about everyone on this thread has lost track of the focus of > PO's claim. As I generally don't consider engaging with PO a useful > activity, I'll just post once, in case any other posters find my > analysis helpful. (Probably this is way too deep in the thread to make > any difference, but hey...) > > The /focus/ of PO's claim is not whether or not the reals always have a > next point; it's that THE REALS ARE COUNTABLE. Agreed. The reals are not countable, and Cantor's elegant proof of this fact has been well known for well over a century. I've certainly seen no refutation of it here. [...] -- Keith Thompson (The_Other_Keith) kst-u@mib.org Will write code for food. void Void(void) { Void(); } /* The recursive call of the void */