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: Mon, 16 Sep 2019 21:38:40 -0700 Organization: None to speak of Lines: 80 Message-ID: References: <87sgoydzm9.fsf@bsb.me.uk> <87blvlequ8.fsf@bsb.me.uk> <4bmdnZCR0LAZOuLAnZ2dnUU7-RfNnZ2d@giganews.com> <4qqdnSmfIroweuLAnZ2dnUU7-cmdnZ2d@giganews.com> <0IidnQ4NqsdLqh3AnZ2dnUU78dfNnZ2d@brightview.co.uk> 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="19321"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19w6V8RGAc8Vk/1AJ+AzQDy" User-Agent: Gnus/5.13 (Gnus v5.13) Emacs/23.1 (gnu/linux) Cancel-Lock: sha1:2uMKxoikQRbRzAiVNXhWwmmredE= sha1:fTr0bBN7NKXCUmGuzadYwtzRqwM= Xref: csiph.com sci.logic:210878 sci.math:458515 comp.theory:19936 comp.ai.philosophy:19237 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 xb 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. -- Keith Thompson (The_Other_Keith) kst-u@mib.org Will write code for food. void Void(void) { Void(); } /* The recursive call of the void */