An uncountable countable set
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From: mueckenh on 18 Sep 2006 10:54 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. 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. > > >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. And the same kind of extrapolation is appropriate if we investigate the infinite, be it the sequence 1/n or the "bijection" N <--> Q. Regards, WM
From: mueckenh on 18 Sep 2006 11:01 Han.deBruijn(a)DTO.TUDelft.NL schrieb: > Mike Kelly wrote: > > > 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, [ ... snip ... ] > > What more evidence do we need, huh? > > The good news is that you are doing wrong only _one_ thing: infinitary > reasoning. You think that completed infinities do exist. Once you stop > thinking this way, everything falls in its place and you will see that > it is quite meaningful to discuss the probability of "a natural" being > divisible by 3. Bravo. That's it. Regards, WM
From: Randy Poe on 18 Sep 2006 11:04 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. Is that your idea of a citation? - Randy
From: stephen on 18 Sep 2006 11:00 Mike Kelly <mk4284(a)bris.ac.uk> wrote: > Tony Orlow wrote: >> But so, you agree that, given certain considerations, the set of even >> naturals can be said to be half the size of the set of naturals? > Of course. >>If so, >> then aren't the conclusions of cardinality not generally true, > Which conclusion? >>if >> equivalent cardinality is taken to be "the same size, in every respect"? > But that is not a conclusion of cardinality. I keep telling you this. > Maybe you're just not capable of listening? Many people have told Tony this since he first started posting. For over a year now he has been told that "size" does not have a formal definition in set theory, and that cardinality is just one possible interpretation of "size". For some reason he refuses to listen. >> That is, I would say card(x)=card(y) implies size(x) *May Be* size(y), >> but not size(x)=size(y). It more or less divides sets into sets which >> are ABOUT the same size ("about" being a pretty vague term). > "Size" being a pretty vague term for infinite sets, more like. What > does size() mean above? > The term "size" *is* ambiguous when talking abount infinite sets. > Cardinality is *an* analogue to size that applies to *all* sets and is > very useful. You're being extremely obstinate in your refusal to accept > that it doesn't claim to be anything more than that. Presumably because > your complaints about set theory are vacuous without this conceit. Your guess as to why he refuses to listen seems a good one. Stephen
From: mueckenh on 18 Sep 2006 11:11
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 nor in mathematics. Regards, WM |