Uncountability of the Rationals?
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From: K_h on 14 Jun 2010 17:09 "Tony Orlow" <tony(a)lightlink.com> wrote in message news:b3dccfb5-1ab9-4e8e-8b00-aa02952820d7(a)x21g2000yqa.googlegroups.com... On Jun 14, 1:32 am, David R Tribble <da...(a)tribble.com> wrote: > Brian Chandler wrote: > > Remember, Tony is trying to posit that real functions are > allowed to operate on non-real infinite values (assuming a > few criteria are met) just as if those infinite value > acted like > real values. The real function f(x) = x+1 works on all > real x, > but Tony claims that it also works on infinite x in > exactly the > same way. > Yes, I do. Tony, I am reading through this enormous mess you have created (this enormous thread on sci.math), and am bewildered. Why are you doing this? You don't seem to have any clear cut easy to understand system; you just dribble out bits and pieces of undefined, or poorly defined, stuff and then constantly argue with everyone else about it. Why don't you clearly list all of your axioms in a formal way (in one post) that people, who are versed in mathematics, can grasp without ambiguity? Take the real function f(x)=x+1. By definition it only applies to real numbers. We could define another function, on the extended reals, f(x)=x+1 that applies to the extended reals. For all practial purposes the function is the same except the extended real numbers (+oo and -oo) give f(+oo)=+oo and f(-oo)=-oo. On a non-standard number line, for example the surreal number line, a function f(x)=x+1 can be defined that applies to all surreal numbers. In that case you can have things like f(w+pi)=w+pi+1. Notice, the extended real number line and the surreal number line are both well defined mathematical objects. The surreal number line can be formulated within the ZF approach or on its own (As Conway does in his book). If you want to do real mathematics then you need to formulate your ideas into a clear and comprehensive system. You can do it within whatever system you want so long as it is a well defined system. And what is all this stuff about set size in your posts? Within ZF there are other ways of defining size but I have not seen a better definition than cardinality. If you think you have something to contribute to the world of mathematics then why not publish it in one of the journals and/or establish a website for it rather than continuously posting here? The enormous arguing you are doing seems to be one huge waste of time. (As for the title of this thread, the rationals are not uncountably infinite.) _
From: Tony Orlow on 14 Jun 2010 17:16 On Jun 14, 9:20 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Tony Orlow <t...(a)lightlink.com> writes: > > On Jun 13, 3:56 pm, David R Tribble <da...(a)tribble.com> wrote: > >> Jesse F. Hughes wrote: > >> >> I know what the function n |-> n^2 means for infinite values. I'm > >> >> asking you know about the function sqrt. What is sqrt(omega)? What > >> >> properties does it satisfy? > > >> Tony Orlow wrote: > >> > (x>1 -> sqrt(x)<x) ^ (omega>1) -> sqrt(omega)<omega > >> > It's the size of a smaller countably infinite set, the set of squares > >> > of naturals. > > >> All this does is state a property of something you call "sqrt(omega)". > > > Yes, it's less than omega, Bigulosity-wise due to ICI. > > >> You left out the parts that > >> a) actually define what sqrt(omega) is, and > > > Suffice it to say it's countably infinite but less than omega > > You keep dodging the issue. > > Is sqrt(omega) really the square root of omega? That is, is it a number > satisfying sqrt(omega) * sqrt(omega) = omega? > > -- > Jesse F. Hughes > > "Do not click any hyperlinks that you do not trust. Type them in the > Address bar yourself." -- Microsoft gives security advice.- Hide quoted text - > > - Show quoted text - Sure. I didn't dodge. Yes. Affirmative. True. Okay? Tony
From: Tony Orlow on 14 Jun 2010 17:21 On Jun 14, 10:46 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 13, 4:55 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 13, 4:37 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > "complete relation", set of all pairs of members from each set, > > > For any sets S and T, we call that S X T. > > Sorry, CORRECTION: > > I meant (S u T) X (S u T). No, S X T would be a relation between S and T, I think by necessity the complete relation, of count |S|*|T| > > Hmm, it seems there should be a common term for that. Perhaps there > is, but it's not coming to my mind now. Maybe it is 'complete > relation', but I don't recall ever reading it. Well S u T should have a size equal to |S|+|T|-|SnT|, if that's what you mean... ;) So, the complete relation of that set with itself would have a "zize" that is the square of that union's "zize". > > MoeBlee TowNee
From: Tony Orlow on 14 Jun 2010 17:22 On Jun 14, 11:52 am, Brian Chandler <imaginator...(a)despammed.com> wrote: > David R Tribble wrote: > > Brian Chandler wrote: > > > Seems to me something has gone wrong here... > > > Okay, before we all start talking at cross purposes here... > > We already are. > > > Virgil wrote: > > >> Real functions do allow infinite quantities either as arguments or > > >> values, so that you are making false statements. > > > Brian Chandler wrote: > > > Really? What does Virgil mean by "real functions"? One might guess > > > (particular if one were as mathematically ignorant as Tony) that he > > > means "real-valued functions". But to make V's statement true, I > > > think, he would just have to mean "real functions" as opposed to > > > "fake functions". Pretty confusing. > > > Yes, okay, Virgil probably meant "actual" functions, as opposed > > to functions (or "formulae") in Tony's sense. And yes, functions > > can certainly have non-real domains and values. > > No, Virgil explicitly says that he means real-valued functions. But it > turns out that when he wrote "do allow", this was actually a typo for > "do not allow". > > > Tony Orlow wrote: > > >> Don't correct me incorrectly. > > Tony managed to read "do allow" as meaning "do allow", and is rightly > annoyed that Virgil is spouting nonsense -- albeit really a typo for > non-nonsense. But UIMM, Virgil felt it appropriate to respond with one > of his usual spews, without bothering to read his own words to see if > they made sense. > > This is not how communication is supposed to work, chaps (and > chapesses should there be such). > > Brian Chandler Thank you, Brian. <3 Tony
From: Tony Orlow on 14 Jun 2010 17:26
On Jun 14, 12:04 pm, Brian Chandler <imaginator...(a)despammed.com> wrote: > FredJeffries wrote: > > On Jun 8, 3:03 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > MoeBlee refers to extrapolating from the finite to > > > the infinite when considering set size. Here are > > > two equally intuitive notions that describe set > > > size for finite sets: > > > > 1. Sets in bijection with each other have the same size. > > > 2. The whole is strictly greater than the part. > > > (paraphrased from Euclid) > > > > The problem is that for (Dedekind-) infinite sets, there > > > is no example of set size that preserves both of these > > > notions, since by definition there exists a bijection > > > between a D-infinite set and one of its proper subsets. > > > > So the best we can do is choose one of these two equally > > > good notions to preserve. Standard Cantorian cardinality > > > rejects the second in favor of the first. But I see no > > > reason that we can't reject the first in favor of the > > > second -- and this would give us TO's Bigulosity. > > This is nonsense, of course. "Standard Cantorian cardinality" (SCC) > doesn't "reject" anything -- but then TP is totally hung up on > "standard theorists" (or whatever they're called this week) > "rejecting" things. Anyway, yes, cardinality is a natural extension of > the "size" comparison of finite sets which preserves the first notion. > But the simplest measure that preserves the second notion is the > subset relation: perfectly sound "standard theorists'" maths, but > inevitably only a partial ordering. Bigulosity simply isn't a theory > in any normal sense -- just a braindump of Tony's musings. > > > Which of the following is larger? > > A = {1, 2, 3} > > B = {1, 4, 5, 7} > > I can use bijections to see that {1, 2} has smaller cardinality that > > {1, 2, 3} without referring to the fact that the first is a subset of > > the second. But I can't find a way to say which of {1, 2, 3} or {1, 4, > > 5, 7} is larger with respect to your second notion without using a > > bijection somewhere. > > Come on... for each of the two finite sets, write the names of the > elements on evenly spaced points around a circle. Join points in the > obvious way to form polygons. To compare the size of two sets, simply > measure the internal angles of their respective polygons, and > whichever has the larger angle is the larger set. > > Brian Chandler- Hide quoted text - > > - Show quoted text - Eat me, Brian. 3< Tony |