An uncountable countable set
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From: Virgil on 25 Sep 2006 22:51 In article <45187409(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <451149ef(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> 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? 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? > > > > There is no "infinite case", there is only a limit case. > > > > Then noon never comes and the vase is never empty, Both are limit cases, which are what actually occurs, given the models. > > > > > If the cases for a finite number of steps are sets of points, so is the > > limit case. > > Correct. There is no measure. > > > > > If the finite cases are sets of segments with specific directions > > determined by their endpoints, the limit case will only contain pairs of > > identical points which do not determine any direction at all, and so is > > ill defined. > > Incorrect. First of all, each segment was defined by a xy-offset pair, > starting at point (0,0). Each such pair denotes, through these relative > coordinates, a length and a direction. So what length and direction are indicated by the pair {(0,0),(0,0)} ?
From: Virgil on 25 Sep 2006 22:58 In article <45187864(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Han de Bruijn wrote: > > Tony Orlow wrote: > > > >> Mike Kelly wrote: > >> > >>> Han de Bruijn wrote: > >>> > >>>> Mike Kelly wrote: > >>>> > >>>>> Han de Bruijn wrote: > >>>>> > >>>>>> Mike Kelly wrote: > >>>>>> > >>>>>>> What the hell are you talking about? Arguing with someone who can't > >>>>>>> speak English is getting aggravating. > >>>>>> > >>>>>> My English is much better than your Dutch. > >>>>> > >>>>> So what? Your English is still too poor for this discussion to be > >>>>> fruitful. > >>>> > >>>> Still don't get the point, huh? > >>>> > >>>> You are lacking even the most elementary form of politeness. It's very > >>>> impolite to cut of a discussion with somebody from a foureign country - > >>>> somebody who is doing his best to communicate with you - only because > >>>> you are obviously superior in expressing your thoughts within your own > >>>> mother's tongue. > >>> > >>> You're a very rude person yourself, Han. I generally don't feel the > >>> need to be civil to those who won't reciprocate. > >> > >> I don't think I have ever found Han to be rude, except when he > >> referred to my "babbling" recently. Ahem. But anyway, while we > >> disagree on the actuality of any infinity, we have the open mind of > >> spirited debate, and feel no need to get nasty. > > > > I may be rude sometimes, but I never get _personal_ by calling somebody > > an "idiot" or a "crank". Tony's "babbling" translates with Euroglot as > > "babbelen" in Dutch, which is a word I can use here in the conversation > > with my collegues without making them very angry (if I say "volgens mij > > babbel je maar wat"). But, of course, I cannot judge the precise impact > > of the word in English. Apologies if it is heavier than I thought. > > Do you have the saying, "Shallow brooks babble, and still waters run > deep"? I figured you picked up the usage from this forum, actually. It's > meant, in English, to mean you aren't making any sense. :) > > > > >> Furthermore, I have never had any trouble understanding what Han is > >> saying, except where he is using some mathematical construct with > >> which I am not familiar. His English is not bad, and blaming your > >> disagreement on his inability to communicate is kind of low. > > > > Thank you very much, Tony, for this sort of defense. > > My pleasure. It seemed like a vacuous excuse. I get pretty sick of those > diversionary tactics. > > > > >> So, let's engage in lively debate, and maintain our civility, while > >> chopping each other's arguments to pieces. Of course, this can only > >> happen if we don't consider our arguments to be part of our anatomy. > >> Otherwise, it gets personal. > >>> > >>>>> You are misinterpreting virtually all my posts. You claim that you're > >>>>> not dishonest so I have to conclude you're simply incapable of > >>>>> comprehending written English. This makes this whole subthread > >>>>> pointless. > >>>> > >>>> I have only this kind of trouble with _you_ and nobody else on the web. > >>> > >>> Really? You've never had anybody else other than me complain that you > >>> misinterpret their posts? I suppose I must have hallucinated dozens of > >>> posts I've seen of just that, then. > >>> > >>> You've never had anyone other than me struggling to understand what the > >>> devil you mean by your broken English? I must have hallucinated, for > >>> example, "A little physics would be no idleness in mathematics", then > >>> :)? > >> > >> Well, that's a difficult type of quote. Han - I wouldn't mind working > >> on exactly how you want to say that in English, if you like. :) > > > > Uhm, since litteraly everybody is complaining ... Let it be an encrypted > > message then :-) > > Well, it seems to me that perhaps you're saying something like, "Those > with their heads in the abstract should keep their feet in the > concrete", though that sounds a little funny. > > Math=Science? Scientists, particularly those in the sciences most dependent on mathematics, tend to think that all mathematics is, or should be, a subservient to their particular fragment of science. Mathematicians know better.
From: Dik T. Winter on 25 Sep 2006 22:58 In article <1159186907.615747.304410(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > > > > Let me ask you to, before I can answer such a question. What > > > > > > is your definition of "number"? (I asked the same from > > > > > > Wolfgang Mueckenheim, but his answer was not satisfactory, > > > > > > also not to himself, I think, because he never answered to > > > > > > questions about it.) > > > > > > > > > > Didn't you read my paper on the physical constraints of numbers? > > > > > > > > Yes, I read it. Mathematically it makes no sense. > > > > > > You did not yet recognize it, perhaps later. But you were not telling > > > the truth above, were you? > > > > Where did I not write the truth? > > Here: (See above) > > > If I considered Dik as one person, I would write {Dik, Virgil, me} for > > > instance. But you are right, there is some ambiguity. > > > > Yes, so it is not a proper definition. And as such, the above to me still > > makes no sense as a proper definition at all. > > There is a natural number which is the largest one ever mentioned or > thought during the lifetime o the universe. It is not properly defined > before the universe ceases and probably also not afterwards. But it is > or will be (depending on the question of determination or not). > Nevertheless, it does not exist yet. We have to live with those > improper objects. So, again, no definition. Where did I not speak the truth? > > > > > 2) or is completely determined by a series of digits. > > > > > > > > Question. A terminating or a non-terminating series? > > > > > > There are only terminating series. There is no infinity in reality and > > > useful mathematics. > > > > Oh. So you state. But 1/3 is a number? > > 1/3 is a number, properly defined, for instance, by the pair of numbers > 1,3 or 2,6 or 3,9 etc. But 0.333... is not properly defined because you > cannot index all positions, Again, you *ignore* the definition of that notation as a decimal number. I state again, that notation has *no* meaning until some meaning has been defined. In mathematics it is defined as the limit of a sequence. If you think that definition is invalid, you should seriously consider all use of limits in mathematics to be invalid. > > What do you mean with "exisiting"? > > Existing is a thing you can use, like the largest known Fermat-prime, > or the theorem of Pythagoras, or a hot dog in your hands. Eh? > > The set of prime numbers is infinite > > and unbounded. The set of known prime numbers is finite and bounded. > > it is finite, but not bound, because it can, and probably will, grow. > That is the same as with the set of natural or real numbers. The set of known prime numbers is bounded. Period. It is a specific set that now and today consists of a fixed number of elements. It may be a different set tomorrow, but that is something different, and again, tomorrow it will be fixed and bounded. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 25 Sep 2006 23:01 In article <1159211074.494116.142040(a)e3g2000cwe.googlegroups.com> Han.deBruijn(a)DTO.TUDelft.NL writes: .... > I'm still flabbergasted why those difficult proofs as for Fermat's Last > Theorem or the Poincare Conjecture are not proved then with the full > power of modern computers. Perhaps because computers can only be used to prove finitely many cases and that FLT and Poincare are not tangible to be reduced to finitely many cases (like 4CT)? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Tony Orlow on 25 Sep 2006 23:12
cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: >> cbrown(a)cbrownsystems.com wrote: >>> Tony Orlow wrote: >>> >>> <snip> >>> >>>> I did give another curve with the same "Tlimit" as the staircase in the >>>> limit, which produced an interesting result, giving weight to the notion >>>> that an infinitesimal is something distinct from 0, whose square is not >>>> distinct from 0. >>> Suppose we let B represent Big'un; then B*1/B = 1, where 1/B is an >>> infinitesimal. Then what you are saying means >>> >>> 1/B = 1/B >>> 1*1/B = 1/B >>> (B*1/B)*1/B = 1/B >>> B*(1/B*1/B) = 1/B >>> B*(1/B^2) = 1/B >>> >>> Since 1/B is infinitesimal, its square is not distinct from 0; so... >>> >>> B*(0) = 1/B >>> 0 = 1/B >>> >>> So 1/B is identical to 0. Where is my error? >>> >>> Cheers - Chas >>> >> Hi Chas - >> >> I haven't been online lately, so please excuse the delay. You haven't >> made any error, except in interpretation of my position. I said that I >> got an interesting result from applying a segment-sequence definition of >> your staircase-and-diagonal comparison, which gave "weight" to the >> notion of an infinitesimal being defined in such a way. > > Well, why doesn't the logic I gave above give obvious "weight" to the > notion that your nilpotent infinitesimals imply that 1/B = 0? Or, since > then B*1/B = B*0, that 1 = 0? > > Doesn't that tend to contradict your "intuition" regarding T-numbers? > > Cheers - Chas > Not really. As I said, there are two ways to consider infinitesimals, which each have their applications. Nilpotent infinitesimals are useful it seems for measure, where an infinitesimal portion of an infinitesimal unit can be ignored in the measure of an infinitesimal portion of the finite whole. In other words, if x is infinite, then in standard measure, 3+1/x=3. If we divide both sides of that equation by x, we get 3/x+1/x^2=3/x. So, in the measure of that infinitesimal portion of the whole, the sub-infinitesimal 1/x^2 can be neglected. Infinitesimal portions have no relative measure. On the other hand, that sub-infinitesimal square of the infinitesimal 1/x can be considered a significant difference, so that 3/x+1/x^2>3/x, just like when infinitesimals are considered as inverses to infinities and as nonzero in some nonstandard unmeasurable sense, we can say 3+1/x>3. Does a hyperbolic space contradict a parabolic one? Tony |