An uncountable countable set
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From: Mike Kelly on 20 Sep 2006 14:45 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: > > Han de Bruijn wrote: > > > Tony Orlow wrote: > > > > > >> Han de Bruijn wrote: > > > > > >>> Precisely! Mathematicians get confused by the idea of a "bijection", > > >>> which is an Equivalence Relation, which in turn is a "generalization" > > >>> of "common equality" (yes: the one in a = b). But the funny thing is > > >>> that EQUALITY HAS NEVER BEEN DEFINED. > > Idiot. Do you know the definition of an equivalence relation? Do you > claim that bijection is not an equivalence relation? > > (I think that was a different crank - here's Tony...) > > > Consider the equally spaced staircase from (0,0) to (1,1), as the number > > of steps increases from 1 without bound. Is it the same as the diagonal > > line? > > What _exactly_ is "it" here? If I consider the set of staircases as the > number of steps increases without bound I get an unending set of > staircases. The only obvious singular object is the set, which is not > anything like a diagonal line (of course I don't think you mean this); > otherwise there are lots of staircases. Well, now you foam at the mouth > a bit... > > > Inductively we can prove that the length of the staircase is 2 at > > every step. Does it really suddenly become sqrt(2) in the infinite case? > > By the measures of point set topology, all points in the staircase > > become indistinguishable in location from the those of the diagonal, so > > by this thinking, all difference has disappeared, and the two objects > > are equal. However, using a segment-sequence topology, staircase n is > > the concatenation of n pairs of segments, denoted by x and y offset, > > described by {0,1/n} {1/n,0}, whereas the corresponding segments of the > > diagonal, between the points on the diagonal where perpendicular lines > > pass through the vertices of the staircase, are of the form > > {sqrt(2)/2n,sqrt(2)/2n}. The fact that the directions of the two curves > > are different at every point explains the difference in length, but this > > distinction cannot be detected by looking at pointwise location alone. > > Blabbley-blobbley. I was reading the Wikipedia article on "Crank > (person)" today. Particularly the bit about cranks' incredible > over-rating of their own abilities. You seriously think you are so much > cleverer than the staff of every maths department in the world that you > alone can notice that every one of these staircases has length 2; you > think mathematicians in general are _that_ stupid? > > Mike Kelly [I think] went all through the stuff about limits I've gone over limits with Tony several times. But the "staircase" example was Chas' (...@cbrownsystems.com). Extremely well done it was too. A shame Tony wasn't able to appreciate it; I learned quite a lot myself. -- mike.
From: imaginatorium on 20 Sep 2006 14:55 Tony Orlow wrote: > imaginatorium(a)despammed.com wrote: > > Tony Orlow wrote: > >> Consider the equally spaced staircase from (0,0) to (1,1), as the number > >> of steps increases from 1 without bound. Is it the same as the diagonal > >> line? > > > > What _exactly_ is "it" here? > > Idiot. What was the object referred to in the previous sentence? Do you > not know how to correlate pronouns to their reference? (Incidentally, I think you mean "referents".) Oh, OK, "it" is "the" staircase. But we are considering this staircase "as" something else varies - so actually a whole family (a sequence) of staircases. > > If I consider the set of staircases as the > > number of steps increases without bound I get an unending set of > > staircases. The only obvious singular object is the set, which is not > > anything like a diagonal line (of course I don't think you mean this); > > otherwise there are lots of staircases. Well, now you foam at the mouth > > a bit... > > I mentioned ONE staircase, in the limit as the number of steps > approaches oo. Don't play dumb. Yes, if you were mathematically capable, I would guess you simply meant the (properly defined) limit of the sequence of staircases. But you don't, because you haven't understood the mathematical notion of a limit at all, and you hope you can conjure up some mysterious-well-it's-kinda-like-a-staircase which is both a staircase, and somehow "in the limit". Your attempts to define this of course rely on undefined notions such as "infinitesimal steps". Standard crank technique: an "infinitesimal" x is an x, except that when it being an x leads to a contradiction you can just ignore it. Really, and truly of course, your "it" is the staircase you get to when you reach the end of the unending sequence of staircases (if that sounds familiar, it probably is). > >> Inductively we can prove that the length of the staircase is 2 at > >> every step. Does it really suddenly become sqrt(2) in the infinite case? Do you have a reference to a textbook that says that anything "suddenly becomes" anything "in the infinite case"? You might like to know that the expression "in the infinite case" really is a pointer to the root of your muddle - it shows that you think that by chunking through the numbers faster and faster one eventually reaches the end, known as "the infinite case". Well, no hope of that muddle ever being cleared up, is there? > I was reading the Wikipedia article on "Crank > > (person)" today. Particularly the bit about cranks' incredible > > over-rating of their own abilities. You seriously think you are so much > > cleverer than the staff of every maths department in the world that you > > alone can notice that every one of these staircases has length 2; you > > think mathematicians in general are _that_ stupid? > > I think I have a nonstandard perspective which is at least equally valid > as the standard transfinitology. Yes, we know you do. Have you read the Wikipedia article on "Crank (person)"? A small quotation: "... the essential defining characteristic of a crank: "No argument or evidence can ever be sufficient to make a crank abandon his belief." We're here for the entertainment, mate. Brian Chandler http://imaginatorium.org
From: Tony Orlow on 20 Sep 2006 15:10 MoeBlee wrote: > Tony Orlow wrote: >> Given that all evens are natural, but only every other natural is even, >> is is logically deducible that the naturals are twice as numerous in any >> range as the evens. > > Define 'twice as numerous' for infinite sets. IFR does that and more, in terms of formulaic relations between sets over a given range. Where the mapping from the naturals to another is f(x)=x/2, the other has twice as many elements over any range that's a multiple of 1, and asymptotically, over the entire real range. > > It's a theorem of set theory is that there is bijection between the set > of natural numbers and the set of even numbers. It's also a theorem of > set theory that every other natural is an even natural in the standard > ordering of naturals. This is not a contradiction. No, but those two statements, combined with the statement that bijection->equivalence, and that a proper subset must have a smaller size than its proper superset, imply that bijection alone cannot be an ***accurate*** measure of set size, when it comes to infinite sets. By "accurate" I mean that any addition of elements, or multiplication or exponentiation of multiple preexisting elements, will result in a larger set, as reflected by this measure. > > MoeBlee > ToeKnee
From: Randy Poe on 20 Sep 2006 15:10 Tony Orlow wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Given that all evens are natural, but only every other natural is even, > > > > Yes, the evens are a proper subset of the naturals. > > > >> is is logically deducible > > > > No, that's not a deduction. There's no theorem or axiom you > > use to make this jump. > > IFR does that and more. I haven't followed this discussion enough to know what "IFR" refers to, but I suspect this is one more case where you've declared it's true because you declare it's true. That of course is not a proof. > Or, I can be a simpleton and refer to relative > set densities. That of course is not a proof. You can't prove this implies "twice as numerous" until you define "twice as numerous". It sounds like by "twice as numerous" you really mean the idea of density, that in some sense we can agree on, the density of the evens on the naturals (a) exists and (b) is 1/2. In other words, you want to "refer to" this defined quantity of 1/2 in order to prove that this defined quantity is 1/2. I won't argue with that, but it doesn't seem particularly useful. > > The problems start when one tries to pin down a > > meaning of "numerous" for such sets. Then the > > way things are no longer follows "the way Tony > > wants them to be". > > Indeed they do, if we make them. Hold onto the concept that less is not > equal, and you'll see why proper subsets are never as large as their > proper supersets. You just made a jump, that "less than" automatically means "is a proper subset of". That's one possible ordering relationship. It's unsatisfactory for set theory because it's a partial ordering. For sets A and B which do not have a subset relation, we can't say A<B, A>B or A=B. They aren't related. > > > > > But I never worried about what you were thinking > > when I learned mathematics, so it's not overly > > bothersome to me that the deductions lead to a > > different conclusion than the desires of some stranger > > named Tony on the internet. > > When you learned transfinitology, had you seriously considered the > mathematical notion of infinity previously? I had. I've never heard of "transfinitology". If you mean had I tried to reason about infinite sets before seeing the proof of countability of the rationals at age 10, the answer is no. > >> Intuition is a wholistic application of logic and > >> association, > > > > No, intuition is guessing in the absence of logic. > > Um, you might want to do more research into the structure and behavior > and laws of mind, before you claim to know what intuition is. I use intuition all the time, but I recognize that it isn't the same as proof, and that it can be wrong. The difficulties I have with you is your insistence that intuition is BETTER than proof, that if the intuition and the proof lead to different conclusions, then the proof is "wrong". When you cling to a belief in the face of overwhelming evidence to the contrary, that is faith-based. It isn't anything like mathematics. - Randy
From: imaginatorium on 20 Sep 2006 15:17
Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> >> Actually infinite(S) <-> E seS A neN index(s)>n > >> >> Potentially infinite(S) <-> A seS index(s)eN ^ A neN E seS index(s)=n. > > > > So this is what you have now ('w' stands for 'N', which stands for?): > > Where do you see 'w'? > > > > > S is actually infinite <-> Es(seS & An(new -> index(s)>n)) > > I would rather not use 'w'. Sick to N. Why not? > > > > > S is potentially infinite <-> As(seS -> (index(s)ew & An(new -> Es(seS > > & index(s) = n)))) > > > > Okay, now I see that your 'actually infinite' is something like what > > set theory would describe as 'S has a member greater than any member of > > w'; and your 'potentially infinite' is what set theory would describe > > as 'S has only finite members but S has a denumerable number of finite > > members'. > > Okay. Never mind the confusion over the letters - there seems to be a big problem here, in that the definition of "potentially-Tinfinite" and "actually-Tinfinite" makes reference to a set N, which I suppose is some sort of large set including lots (all?) natural numbers, of one flavour or another. How then would we ask whether this set N is either sort of Tinfinite? Brian Chandler http://imaginatorium.org |