An uncountable countable set
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From: stephen on 20 Sep 2006 11:00 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > stephen(a)nomail.com wrote: >> Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: >> >>>Try to understand what equality means. >> >> That does not seem to be an answer. Your position is >> that "represenation is number". According to you there >> is no difference between a number and its representation. >> "3/2" is a representation. It is different than "6/4". >> Or does your definition of "equality" somehow allow for >> different strings to be equal? > Read the response by WM. I cannot improve on it. > Han de Bruijn Is sqrt(2) a number? Stephen
From: Tony Orlow on 20 Sep 2006 11:08 Han de Bruijn wrote: > Tony Orlow wrote: > >> Han, if I prove inductively, say, that 2^x>2*x for all x>2, do you >> find it objectionable to say that this also applies to any infinite >> value, if such a thing existed, given that any infinite value would be >> greater than any finite value, and therefore greater than 2? > > Give me one reason, Tony, why I would find such a theorem interesting > in the first place. I'd prefer the ultimate terseness in mathematics, > especially if it comes to infinities. Okay, but it's not that particular theorem that is of interest, but the system of theorems this extension of induction makes possible. If we can say that such inductive arguments hold in the infinite case, and within any range up to n one set has size 2n (multiples of 1/2) and the other x^2 (logs base 2 of naturals), then proving that n>2 -> n^2>2n proves the second to be bigger than the first, for infinite n, since they are larger than 2. This gives us a nice exact way of comparing infinite sets over an infinite range, free from "limit ordinals" and other nonsense. This allows us to distinguish between the sizes of the naturals vs. the evens, and even detect the addition or removal of a single element from an infinite set. Hope that made sense. > >> I think if Wolfgang and Han were offered a more sensible treatment of >> the infinite case, they might find it more palatable. > > Affirmative. That's nice to know. :) > > Han de Bruijn >
From: Tony Orlow on 20 Sep 2006 11:10 Han de Bruijn wrote: > Tony Orlow wrote: > >> The agreement that I think Han and I came to in "Calculus XOR >> Probability" was that such probabilities are infinitesimal. > > Affirmative. Sometimes people _do_ agree, even in 'sci.math'. > > Han de Bruijn > It's not surprising when they agree that the standard is correct. It's more interesting when people arguing against the standard can agree on the alternative implications of a problem. That's kind of exciting. Tony
From: imaginatorium on 20 Sep 2006 11:14 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 (that's the mathematical term, with the normal mathematical meaning, not any of your waffle) with you - to no effect, obviously, since overawed by the power of your own brain, you don't have time to understand what the term means. You seem to have some notion of your own (ice pose a "Tlimit"), which is some property of a sequence such that it embodies any property of all members of the sequence. Trouble is, in the end this is just the sequence itself. You can't give two _different_ sequences that have the same Tlimit. So a Tlimit captures no generalisation at all. (You won't know this, but in a sense mathematics is precisely the business of capturing generalisations.) Brian Chandler http://imaginatorium.org
From: imaginatorium on 20 Sep 2006 11:25
Tony Orlow wrote: > Virgil wrote: <snip> > > Which axioms in which mathematical systems does TO think have not been > > judicially chosen? > > The axiom of choice, for one. That's optional even in your system. Would you care to quantify who the "you" refers to in "your system"? Do you imagine that the four horsemen of the apocalypse [g] have constructed our own "system"? Brian Chandler http://imaginatorium.org |