An uncountable countable set
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From: Mike Kelly on 18 Sep 2006 09:42 Han de Bruijn wrote: > Mike Kelly wrote: > > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > >>What I meant to express is that you are about to be parrotting > >>mainstream arguments, without adding to it much thoughts of > >>yourself. And that is quite senseless because we have gone > >>through all this already. > > > > Yes, and your position was utterly ripped apart in the very first > > response to you by David C. Ullrich, and then by several others. You > > were unable to defend your claim. So, why repeat it as though it were > > in any way valid? > > That's because I did a _honest attempt_ to fit my position into the > framework of nonstandard analysis (Robinson's theory), which failed. Your claim was that *standard* set theory + calculus contradicts *standard* probability theory. This is untrue. Do you admit it? > Of course it failed. Mainstream mathematicians have never understood > how infinitesimals work. NSA is part of mainstream mathematics. What mathematics have never understood is why some people are so enarmoured of vigorous handwaving as a form of mathematical argument. -- mike.
From: Mike Kelly on 18 Sep 2006 09:46 Han de Bruijn wrote: > Mike Kelly wrote: > > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > >>Mike Kelly wrote: > >> > >>>Given that any second-year student of probability theory knows that > >>>there are no uniform distributions over countable sample spaces, [ ... ] > >> > >>This "given" is most disturbing. Mainstream mathematics is so certain > >>about its own right that no sensible debate is possible. > > > > Please stop snipping so much context. It is dishonest. > > Why ?! Because it distorts the meaning to remove necessary context. >Everybody should be able to look up the rest in the first place. Why should they? Don't remove necessary context. > I'm just snipping the parts that don't belong to the subject "Given ... > blah .." Nothing dishonest, just sizing down the universe of discourse. Huh. No, you're quoting me out of context, repeatedly. Looking at your antics in other threads you appear to make quite a habit of this so I don't suppose I'll be able to disuade you. -- mike.
From: Mike Kelly on 18 Sep 2006 09:47 Han de Bruijn wrote: > Mike Kelly wrote [ dishonestly snipping again ]: > > > I haven't questioned your ability to use the calculus. > > Now use your common sense. Why would somebody who clearly has _some_ > mathematical abilities be a complete crank if it comes to a subject > which is somewhat different from calculus, but still is mathematics? Uh, because competence at applying one area of mathematics doesn't automatically make you competent in every other area of mathematics? -- mike.
From: Mike Kelly on 18 Sep 2006 09:51 Han de Bruijn wrote: > Mike Kelly wrote [ OK, let's keep the quotes intact this time ]: > > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > >>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? > > > > Given that this is a *theorem* of probability theory I am mystified as > > why this is evidence that I don't understand probability. Do you have > > some alternative probability theory? > > The problem is the layer below Probability theory: Set theory. You say > it correctly here: So.. are you still claiming I don't know understand probability? Did you ever actually mean it or was it just a stupid thing you felt better for saying? > >>The good news is that you are doing wrong only _one_ thing: infinitary > >>reasoning. You think that completed infinities do exist. > > > > If you don't accept the existence of a set of natural numbers then you > > don't accept the set theory that probability theory is based upon and > > you haven't suggested an alternative. Indeed, it seems somewhat odd to > > complain about the conclusion of a theorem discussing an object you > > don't accept even exists. > > The infinitary part of set theory that underpins probability theory is > IMHO the only problem. Then it's stupid to attack probability theory as it obfuscates what your real disagreement is. > >>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. > > > > 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. > > The core of the matter is that THE naturals can only exist as a set of > consecutive naturals 1 thru n where n is large and undefined. Why? > Any such set is equipped with a uniform distribution. And hence "THE" naturals. Such a set doesn't contain all naturals, so in what sense is it "THE" naturals? > This does not say that meaningful answers (i.e. independent of n) can > always be obtained. But: mainstream mathematics HAS found a way out of > al this. It's called the theory of "natural densities" or some such .. > Why not substitute? But this belongs to another thread: > > http://groups.google.nl/group/sci.math/msg/225dca8f63d0d6ae?hl=en& Not changed much, have you? -- mike.
From: Mike Kelly on 18 Sep 2006 09:54
Han de Bruijn wrote: > Mike Kelly wrote: > > I want to snip all this but Mike doesn't like it: > > > 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. 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. > > So why bother to argue against individual theorems? You don't accept > > *any* of probability theory. > > > > 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. > > End of virtual snipping. > > > If you refuse the idea of infinite sets, what does it mean to you to > > say a function has domain and range R? > > I don't really refuse the idea of infinite sets. (How else could I do > my calculus stuff ?) I only refuse the idea that the infinite can be > something which is essentially different from the _finite_. Then you refuse the idea of infinite sets. Infinite *means* "not finite". To claim it means "finite" is just boneheaded. > In short: > infinity is just finity in disguise. No. > That's why I have no problem with limits. But I _have_ a problem with > aleph_0. No, I have no problem with R, because I can add uncertainity > to its members, and then the reals become just floating point numbers > in the PC on my desk. Reals are very handsome idealizations of floats. > > But aleph_0 is not an idealization and it's not even handsome ... Bully for you. -- mike. |