Cantor Confusion
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From: Virgil on 10 Dec 2006 19:08 In article <457c60d0(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> stephen(a)nomail.com wrote: > >>> Han.deBruijn(a)dto.tudelft.nl wrote: > >>>> stephen(a)nomail.com schreef: > >>> Do you or do you not wish to abolish any mathematics > >>> that involves infinity? If you are perfectly content > >>> to let others freely explore whatever they wish, then > >>> why are you so aggressive? > >>> > >> I believe Han agreed that, if he saw a more satisfying treatment of > >> infinite sets, he'd be open to it. Of course, he's been rather resistant > >> to my alternatives, but it seems everyone is, for one reason or another. > > > > Because they're impossible for anybody but you to understand and of no > > apparent utility anyhow. > > > > That's in the mind of the beholder - you. That is a very plural "you". It seems to cover everone who has an opinion at excepting only TO himself. > > >> In any case, I think his objections are specific enough to warrant > >> some attention. "Calculus XOR Probability" was about the loss of > >> additive probability measure, when you have an infinite set of equally > >> likely possibilities, as a result of the notion of aleph_0 elements, and > >> its standard inverse, if there is such a thing, of 0% probability each. > >> No sum of 0's can be anything but 0. Is it unreasonable to want to > >> preserve additive measure within probability over an infinite set? I > >> don't think so, and the answer to that issue was obviously to allow some > >> infinitesimal probability for each natural. > > > > Doesn't work. Even in systems with infinitesimals, a countably additive > > union of sets with measure zero has measure zero. There is no uniform > > probability distribution over the natural numbers. Even if you "allow" > > infinitesimals. > > > > Measure zero is not the same thing as infinitesimal measure. Pay attention. > > >> In that case it can easily follow that the probability the n/3 e N is 1/3. > > > > Only by vigorous handwaving by people who really have no idea what they > > are talking about. > > > > Or by simple logic. Not if that 'simple logic' respects definitions. > > >> Set theory "disagrees". Interpret that as you wish. > > > > Measure theory and probability theory do too, more to the point. > > > > Great. Contradictions within mathematics. What contradictions "within" anything? It is just that mathematics says that given certain definitions as in probability theory there is no probability distribution on elements of a countable set in which each member has the same probability as every other member. > Wouldn't Hilbert be proud. Hilbert would certainly agree that the probability theory, as defined, does not do what its definitions prohibit it from doing.
From: Virgil on 10 Dec 2006 19:14 In article <457c62c7(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > Dik T. Winter schrieb: > > > > > >> > > Pray reread what I wrote: "the nodes can be made to represent numbers > >> > > in > >> > > your tree". That is an easy exercise, I even did show it. The same > >> > > for > >> > > the edges, I did show that too. So actually the nodes and edges also > >> > > represent numbers in some way. > >> > > >> > In the same way as the first few digits of a real number represents a > >> > number. 3.1, 3.14, and so on represent numbers in some way. But that is > >> > not at all important or interesting for the tree argument. > >> > >> So why did you state that I erronously believed that nodes represent > >> numbers? > > > > Because you erroneously did. (The real numbers in the tree are all > > represented by infinite paths like > > 3.1000... > > 3.14000... > > etc.) > > > > Correct, though I might restrict that to fractions. One needs to choose > an interpretation of the tree and stick with it. > > One can view each infinite path as a real in [0,1), in which case each > level of the tree represents a bit position, and each edge/node on that > level represents a choice of 0 or 1 for that bit position. > > If the nodes represent values, you have a quite different situation. As there is an easy and obvious bijection between nodes and node-rooted finite sub-paths, in which each node is paired with the finite sub-path connecting it to the root node, the situations are identical.
From: Lester Zick on 10 Dec 2006 19:26 On Sun, 10 Dec 2006 17:39:57 +0000 (UTC), stephen(a)nomail.com wrote: >Han.deBruijn(a)dto.tudelft.nl wrote: >> step...(a)nomail.com schreef: > >>> Do you or do you not wish to abolish any mathematics >>> that involves infinity? If you are perfectly content >>> to let others freely explore whatever they wish, then >>> why are you so aggressive? > >> _What_ infinity. That's the question. Mainstream mathematics has mixed >> up infinity so much that it's not a sensible notion anymore. > >> Han de Bruijn > >What is mixed up about infinity? You can't define it. > Care to cite an example >where mainstream mathematics has mixed up infinity? Define infinity. > And >shouldn't people be free to explore whatever infinities >they wish? Sure if they can't define what they're exploring. > You were the one complaining that about >your freedoms being restricted, yet you seem perfectly >content to deny those freedoms to others. ~v~~
From: stephen on 10 Dec 2006 20:29 Han.deBruijn(a)dto.tudelft.nl wrote: > Virgil schreef: >> In article <1165695789.274304.75780(a)80g2000cwy.googlegroups.com>, >> Han.deBruijn(a)DTO.TUDelft.NL wrote: >> >> > stephen(a)nomail.com schreef: >> > >> > > functions, etc. as all of those things can be modelled with set theory. >> > >> > The topic of functions has been handled separately on my web page: >> > >> > http://hdebruijn.soo.dto.tudelft.nl/www/grondig/natural.htm#fd >> > >> > In a nutshell: the mainstream definition is narrow-minded because the >> > whole notion of _TIME_ is lacking. >> >> Position, velocity and acceleration are specifically expressed as >> functions of time, so I have no idea of what HdB is talking about. > Huh, no. Let f(x) = 2.x then time is involved with multiplying x by 2. That is your idea of time? And you think that cannot be modelled in set theory? You are becoming increasingly irrational. Stephen
From: Dik T. Winter on 10 Dec 2006 20:55
In article <1165701092.516484.245360(a)j44g2000cwa.googlegroups.com> Han.deBruijn(a)DTO.TUDelft.NL writes: > Really? Then you could have joined me in those _lonely_ threads about > Chebyshev and stuff, and solve some real puzzles. There are plenty of > the kind. Yeah, everybody here has ample time at his hands to solve your puzzles. Why you always expect instant answers for your problems escapes me. What did the mathematicians at TU Delft answer when you posed your questions to them? (Yes, there are mathematicians there.) -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |