An uncountable countable set
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From: Mike Kelly on 15 Sep 2006 09:47 mueckenh(a)rz.fh-augsburg.de wrote: > Mike Kelly schrieb: > > > Han de Bruijn wrote: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > David R Tribble schrieb: > > > > > > > >>Tony Orlow wrote: > > > >> > > > >>>Wolfgang's and my position is that N is unbounded but finite, > > > >> > > > >>"Unbounded but finite" is a contradiction, meaning "not finite but > > > >>finite". I'm sure you and Wolfgang think this double-think makes > > > >>sense, but the rest of us don't. > > > > > > > > Your position only reflects the miseducation in mathematics during the > > > > last decades. > > > > > > > > Actual or finished infinity is a contradicton. Surpassed infinity is a > > > > contradiction. > > > > > > > > Unbounded but finite is mathematical reality. Think of the set of all > > > > natural numbers which have been realized by writing down these numbers. > > > > Think of the set of known prime numbers. Think of the set of written > > > > novels. Think of the set of postings. > > > > > > > > These sets are unbounded because they can be extended without end. > > > > Nevertheless they are always finite. > > > > > > Sorry for jumping in so late. But VM is quite right, of course. We have > > > encoutered utterly absurd consequences of thinking otherwise, like the > > > mainstream "theorem" that the probability of a natural being a multiple > > > of 3 doesn't exist. While the obvious truth is that it is equal to 1/3 . > > > > > > This topic has been discussed at length in a thread called "Calculus XOR > > > Probability". Let Google be your friend, eventually. > > > > > > Han de Bruijn > > > > So you still don't know what "probability" means. How predictable. > > Sorry, would you agree with me that the probability to take a natural > number divisible by 3 from a box contaning the numbers from 1 to 9 is > just 1/3? If you mean "the probability of selecting a number divisible by 3 from a uniform probability distribution on the naturals from 1 to 9 is just 1/3" then yes. > Would this probability change if the box contained the numbers from 1 > to 900 or to 90000? > Would it significantly change if the box contained the numbers from 1 > to 100000? I suppose not. So here we have some idea that selecting a ball from a box that contain a finite multiple of 3 number of balls numbered with consecutive naturals has a 1/3 probability of selecting a ball divisible by 3. Fine. > If you agree with me up to that point, Give or take imprecise language. >then it is clear Clear to who? It's not clear to me that providing finite examples then saying "obviously this holds for infinite cases too" without any justification whatsoever should be at all convincing to anyone. >that the chance > remains the same if we have all natural numbers in that box (provided, > "all natural numbers" is a meaningful expression). Provided that we can "have all natural numbers in a box" and then select uniformly at random from them, you mean? Well, we can't do that. If you are proposing a physical experiment then I think you will have trouble collecting a ball for each natural number, much less fitting them all into a box. If you are discussing probability theory then there is provably no uniform distribution over a countable sample space. Loosely, this is because probabilities in probability theory are real numbers in the interval [0,1] and sum to 1 for a distribution. The sum of a countable infinity of 0s is 0 and the sum of a countably infinity of a constant non-zero real number is not convergent and certainly does not sum to 1. Thus no uniform countable probability distribution exist. Thus the phrase "a randomly selected natural" is meaningless in probability theory if it is taken to imply a uniform distribution over the naturals. I have a feeling you know this already, however! What else might you be talking about if not probability theory or a physical experiment, I wonder? > The mathematics of > the infinite can only be derived from the mathematics of the finite > (because nobody has an idea what "the infinite" is). Don't extrapolate from yourself so harshly! What's interesting to me here is that your statement seems rather Platonic in that it asserts the existence of some "the infinite" and "the finite" the mathematics of one of which can be observed directly by humans and one of which cannot. Yet earlier you were arguing against a literal interpretation of the "existence" of numbers. What's changed? >Otherwise the > limit of the sequence 1/n might be 100. Nobody could prove that false. Babble. -- mike.
From: stephen on 15 Sep 2006 10:02 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > Mike Kelly wrote: >> Han de Bruijn wrote: >> >> Plagiarism? I don't get it. Who is plagiarising what? > "Your" would-be arguments against mine are not really yours. They are > just a _plagiary_ of well-known "arguments" employed by the mainstream > mathematics community. That is pretty pathetic Han. So any mainstream argument is plagiarism? That is a convenient way to dismiss anyone who disagrees with you. What is wrong with you? What is the source of your hostility towards mathematics? Stephen
From: Mike Kelly on 15 Sep 2006 10:07 Han de Bruijn wrote: > Mike Kelly wrote: > > > Han de Bruijn wrote: > > > > Plagiarism? I don't get it. Who is plagiarising what? > > "Your" would-be arguments against mine are not really yours. They are > just a _plagiary_ of well-known "arguments" employed by the mainstream > mathematics community. Where did I claim to have invented probability theory or theorems of it? Is it necessary to invent probability theory yourself from scratch to have an understanding of it? The word 'plagiary' implies misappropriation, which simply is not taking place here. I am actually astounded that you are claiming that employing a mathematical argument that is not your own invention is plagiarism. Perhaps you are simply unaware of the meaning and connotations of the word. Plagiarism is dishonest and in many cases criminal. A fairly hefty accusation. You claimed that you have a very much better understanding of probability than me. Since you know nothing of my knowledge of probability other than that I disagree that it is meaningful to discuss the probability of "a natural" being divisible by 3, it seemed that either you must have a vastly inflated opinion of your own understanding (to the point where you are the foremost probability theorist in the world) or that anyone who holds my point of view is obviously completely unknowledgable about probability theory. Given that any second-year student of probability theory knows that there are no uniform distributions over countable sample spaces, you must be claiming the former. I would certainly call that unabashed egotism and from everything I've read of your ideas about probability it seems entirely unjustified. How absurd accusations of plagiarism provide a solid retort to this, I do not know. Also, please stop snipping context prematurely. It's making it increasingly dififcult to work out what the Dickens you are babbling about. Finally, please stop with the scare quotes. They make you look like a "tool". -- mike.
From: Virgil on 15 Sep 2006 14:33 In article <1158310937.244396.84450(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David R Tribble schrieb: > > > Tony Orlow wrote: > > > Wolfgang's and my position is that N is unbounded but finite, > > > > "Unbounded but finite" is a contradiction, meaning "not finite but > > finite". I'm sure you and Wolfgang think this double-think makes > > sense, but the rest of us don't. > > Your position only reflects the miseducation in mathematics during the > last decades. People objecting to such idiocies as "not bounded but bounded" has been occcuring since before Euclid, so how many "decades" is "Mueckenh" including? > > Actual or finished infinity is a contradicton. What does, or does not, occur in "Mueckenh"'s philosophy is, fortunately, of no interest to anyone but "Mueckenh".
From: Virgil on 15 Sep 2006 14:39
In article <1158311132.535310.110700(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > III is a representation of 3. > > > > "III" and "3" are both numerals representing the same number, but > > neither is anything more that a representation, neither is the number > > itself. > > What is the number denoted by 3? Some cloudy idea in the platonic > universe or heaven? The number 3 is present in this line. It is "3" and > it is "III" and it is "{a,b,c}". Everything else is purest matheologial > rubbish. Then, according to "Mueckenh", 3 = III = {a,b,c}. According to everyone else, 3 and III are names for a property of the set {a,b,c}. |