An uncountable countable set
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From: Mike Kelly on 18 Sep 2006 16:42 mueckenh(a)rz.fh-augsburg.de wrote: > Mike Kelly schrieb: > > > > > 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! > > I talked to a lot of first-rate mathematicians. This in parentheses is > a qoute from many of them. > > > > 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? > > Nothing has changed. There is no complete set of natural numbers. Why not? > Any set that can be established is a finite set. Why? > Hence, the probability to select a number divisible by 3 is 1/3 or very very close to 1/3. >From finite sets of consecutive naturals when selecting with a uniform distribution, sure. But you don't accept that there is a set of "all" natural numbers so what does it mean to you to select at random from "all" naturals? > > >Otherwise the > > > limit of the sequence 1/n might be 100. Nobody could prove that false. > > > > Babble. > > No. Just this is the point! The series 1 + 1/2 + 1/4 + ... is 2 (or at > least as close to 2 as we like), not by definition and not by any > axiom, but by rational thought. Prove that to be the case without using any definition of what a series is and without any axioms. -- mike.
From: Mike Kelly on 18 Sep 2006 16:44 mueckenh(a)rz.fh-augsburg.de wrote: > Mike Kelly schrieb: > > > It is meaningful to say that a natural drawn uniformly at random from a > > set of consecutive naturals 1 thru 3n has a 1/3 probabaility of being > > divisible by 3. Nobody disputes this. But talking about the probability > > of "a natural" being divisible by 3 implies a uniform distribution over > > the naturals. Such a thing does not exist. > > Talking about sinx / x for x --> 0 does not imply the existence of sin0 > / 0. Neither does the result 1/3 imply the distribution for a realy > infinite set f naturals. There is no real (actual, finished) infinity, > neither in physics Ok. >nor in mathematics. Why? -- mike.
From: Mike Kelly on 18 Sep 2006 16:53 mueckenh(a)rz.fh-augsburg.de wrote: > Mike Kelly schrieb: > > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > Mike Kelly wrote: > > > > > > > [ ... snip ... ] 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. > > > > > > It may be not clear to any mathematician, but it is clear to any > > > scientist. The reason is that infinities do not really exist. > > > They only exist as an attempt to make the "very large" rigorous > > > in some sense. The moment you forget this, you get into trouble. > > > > But we are discussing whether there exists a uniform distribution over > > the naturals. > > Please make just the experiment. Choose at random 30 natural numbers > from the whole set N. What is the result? How many of these 30 numbers > are in fact divisible by 3? (In case you have problems with large > numbers: It is easy to check the divisibility of the number by checking > the divisibility of the sum of its decimal digits.) > > Now, it there a distribution lacking, or is the complete set of natural > numbers lacking? Neither. There is a lack of distribution *for the complete set*. > > If you don't think this claim means anything at all then > > why do you dispute it? If you reject the existence of the set of > > natural numbers then you reject the set theory probability is based on. > > In order to calculate probability we do not need set theory. Pascal and > Fermat, for instance, did it without set theory very well. But they did not use a rigorous probability theory. And a rigorous theory becomes a necessity when dealing with probabilities and statistics beyond the trivial. >Your result > shows only that set theory is not useful in any branch of useful > mathematics. Hard to take you seriously when you say this. If it isn't useful, why is it so widespread? Conspiracy? Force of habit? > > So why bother to argue against individual theorems? You don't accept > > *any* of probability theory. > > We have a better probability theory. Oh really? Where can I read about it? What are its axioms? > > It seem your argument is based on the idea that infinites do not exist > > in physical reality. But mathematics is abstract, so this seems an > > absurd objection. > > > > If you refuse the idea of infinite sets, what does it mean to you to > > say a function has domain and range R? > > As an argument you can choose any real which you really can choose. > > See the experiment above. You don't really believe that you can choose > a natural from the whole set N, do you? Sure I can. I choose 7. But I didn't choose it uniformly at random from all naturals. >But if so, what then is N god > for in probability theory (and elsewhere)? Rigor. -- mike.
From: David R Tribble on 18 Sep 2006 19:45 David R Tribble schrieb: >> Yes, I can see now that these are all finite sets. >> >> And which are proper subsets of infinite sets. The set of all naturals >> that have been written now, for example. Obviously it's an ever >> growing set as time goes on, and will never contain the entire set >> of naturals that are possible. So it's simply a finite subset of N, >> and always will be. >> >> Somehow you are using this fact to "prove" that N can't exist, perhaps >> employing some marvelous mathematical logic that has not been >> tainted by mainstream teachings. You show several finite sets. >> How do they prove anything about infinite sets? > mueckenh wrote: > "a reasonable way to make this conform to a platonistic point of view > is to look at the universe of all sets not as a fixed entity but as an > entity capable of 'growing', i.e. we are able to 'produce' bigger and > bigger sets" [A.A. FRAENKEL, Y. BAR-HILLEL, A. Levy: Foundations of Set > Theory, 2nd ed., North Holland, Amsterdam (1984) p. 118]. Is there a sentence that follows that one, maybe about points of view other than platonistic? > Why should simple infinite sets exist in another way? Just because > there is an axiom which cannot be satisfied like the axiom that there > be a straight bent line? I assume you're talking about there being no set that satisfies the Axiom of Infinity. Why can't there be such a set?
From: Virgil on 19 Sep 2006 01:21
In article <1158591295.350485.163410(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Nothing has changed. There is no complete set of natural numbers. Any > set that can be established is a finite set. Hence, the probability to > select a number divisible by 3 is 1/3 or very very close to 1/3. That presumes that the allegedly finite set of naturals that can be constructed is nearly uniform with respect to divisibility by 3 at least, and probably by other numbers as well. What is the justification for this assumption? |