Cantor Confusion
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From: Dik T. Winter on 13 Oct 2006 09:52 In article <ddeb9$452e55fe$82a1e228$16456(a)news1.tudelft.nl> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > Dik T. Winter wrote: > > > In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > ... > > > It is not > > > contradictory to say that in a finite set of numbers there need not be > > > a largest. > > > > It contradicts the definition of "finite set". But I know that you are > > not interested in definitions. > > Set Theory is simply not very useful. Oh. So you think that Banach spaces are not very useful? You think that a book like "The Algebraic Eigenvalue Problem" is not very useful? You may note that both are heavily based on set theory. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 13 Oct 2006 09:55 In article <452e59c2(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > Dik T. Winter wrote: > > In article <452d140b(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > ... > > > I don't understand this definition either. You write lim{n=1 .. oo}. Is > > > that supposed to be a sum over that range, > > > > Why should it be a sum? > > > > > or do you mean lim(n->oo)? > > > > What is the difference? > > > > > Does 1 belong there? > > > > Why not? > > It serves no purpose but to make the limit look like a sum over a range. > Is lim{n=1 .. oo} any different from lim{n=100 .. oo} or lim{n=-1 .. > oo}? Is that to specify that you are approaching oo from the left? It is just a difference in notation. > And what are you saying about this set? That it cannot exist because > n+1=10n for n=aleph_0? I hope not. Besides, is that even relevant to the > problem? Re-read what I wrote. There is no standard definition about the limit of a sequence of sets. So I gave a definition, and within that definition the limits do exist. But, by that very definition, lim{n -> oo} {n + 1, ..., 10n} = {} -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 13 Oct 2006 10:04 In article <452e5cb8(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > Dik T. Winter wrote: .... > > I define it as a string of digits and it does not represent a number. It > > is only when you give proper definitions of what strings extending > > infinitely far away to the left represent, that you can talk about what > > it represents. In common mathematics there is no such definition. > > When Peano defines the natural numbers, does he talk about what they > represent, or only how they are generated? I do not know what Peano was thinking, but when you take the Peano axioms, they only define an infinite sequence of objacts, that are conventionally called the natural numbers. Once you have gotten those objects you can define arithmetic on them (using the axioms). Once you have defined the arithmetic you can think about representations, and define representations (of which the decimal representation is only one variant). > > That infinite strings to the right define real numbers is entirely due to > > the *definition* of real numbers. And that infinite strings to the left, > > within the theory of p-adics, have specified meaning is entirely due to the > > *definition* of p-adics. (And I may note that in the p-adics there is *no* > > definition for infinite strings to the right.) > > Uh, isn't that what the p-adics define? No. Within the p-adics there are definitions for strings infinitely long at the *left*, not at the *right*. > Or, are you saying there is no > quantity associated with any given p-adic, even though there is order > and arithmetic within the system? Please define "quantity", untill I know what you mean I can give no answer. > > In a similar way, ...111 represents a number in the n-adics. The > > reason is that the sequence 1, 11, 111, ... converges. And so that > > number is 1/(1-n) in the n-adics. Again, with precise definitions about > > what convergence does mean. > > All that aside, each such string has a definite successor and > predecessor, and can be ordered, so it acts as a number system. It acts as. Indeed. But with the p-adic numbers there is no consistent ordering possible. But you should not focus on the strings, they are just representations of the objects. But in the p-adics there is no immediate definition of the successor and predecessor functions. So you have to define them (well, +1 and -1 will do well enough). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: briggs on 13 Oct 2006 10:46 In article <1160703256.468470.108180(a)b28g2000cwb.googlegroups.com>, cbrown(a)cbrownsystems.com writes: > briggs(a)encompasserve.org wrote: >> In article <1160591832.201554.78960(a)c28g2000cwb.googlegroups.com>, cbrown(a)cbrownsystems.com writes: >> > briggs(a)encompasserve.org wrote: >> >> In article <1160546562.540946.205860(a)e3g2000cwe.googlegroups.com>, cbrown(a)cbrownsystems.com writes: >> >> > Dik T. Winter wrote: >> >> >> In article <virgil-372F10.17374709102006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: >> >> >> > In article <J6w6LC.9rL(a)cwi.nl>, "Dik T. Winter" <Dik.Winter(a)cwi.nl> >> >> >> > wrote: >> >> >> > > In article <virgil-9E9CC6.02103209102006(a)comcast.dca.giganews.com> Virgil >> >> >> > > <virgil(a)comcast.net> writes: >> >> >> > > > > Dik T. Winter wrote: >> >> >> > > ... >> >> >> > > > > > The balls in vase problem suffers because the problem is not >> >> >> > > > > > well-defined. Most people in the discussion assume some implicit >> >> >> > > > > > definitions, well that does not work as other people assume other >> >> >> > > > > > definitions. How do you *define* the number of balls at noon? >> >> >> > > ... >> >> >> > > > How about the following model: >> >> >> > > >> >> >> > > And you also start with definitions, or a model. I did *not* state that >> >> >> > > it was difficult to define, or to make a model. But without such a >> >> >> > > definition or model we are in limbo. I think other (consistent) definitions >> >> >> > > or models are possible, giving a different outcome. >> >> >> > >> >> >> > Can you suggest one? One that does not ignore the numbering on the balls >> >> >> > as some others have tried to do. >> >> >> >> >> >> That does not matter, nor is that the problem. You gave a model where you >> >> >> find 0 as answer. I only state that I think there are also models where >> >> >> that is not the answer. Why is a limit of the number of balls over time >> >> >> not an answer? >> >> >> >> >> >> Let's give a simpler problem. At step 1 you add ball 1. At step n you >> >> >> remove ball n-1 and add ball n (simultaneously, I presume). >> >> > >> >> > When you say "at step n", do you have some particular time t associated >> >> > with that step? >> >> >> >> That's somewhat irrelevant. What matters is not what numeric time t is >> >> associated with each step. What matters is the [partial] ordering on >> >> the steps. Associating a numeric time t with each step is a way to >> >> ensure a total ordering. But that's more than we need. >> > >> > What time ensures in this problem is that the notion "after all steps >> > have completed, the state of the vase is..." can be well-defined from >> > the problem statement. >> >> Delusions of physicality. >> >> If you want to define the state of the vase after all steps have completed >> all that is neccessary is to define the state of the vase after all >> steps have completed. > I agree. Where is it stated in the following problem: > > "Suppose V = {V_n} is a sequence of subsets of the natural numbers. > What is lim n->oo V_n?" > > that the definition of lim n->oo V_n is to be defined using pointwise > convergence? > > Now, if I had stated, > > "Suppose V = {V_n} is a sequence of subsets of the natural numbers. > What is lim n->oo V_n, where the limit is defined using pointwise > convergence?" > > then the problem would be well-defined. Without the addition, it is not > well-defined. > > The addition of time in the problem is not a "delusion of physicality"; > it is a sufficient piece of information that translates the problem > from being of the first form above (not well defined) to being of the > second form (well defined). [Summary: I agree with you on everything of substance. The rest is matters of opinion, interpretation and, perhaps, miscommunication] It suggests some implicit assumptions, not all of which may be immediately obvious. To you that's compact clarification. To me that's a delusion of physicality. > (At least, as long as we consider that the proposition "if a ball is > removed at some time t0 and not replaced at any t > t0, then it is not > in the vase at any time t > t0" to be implicit; which I do). > > Of course, one could simply stipulate the conditions of convergence as > above; and that is generally what is done. But all this is to say that > without /some/ such definition given explicitly in the problem > statement, the problem is not well-defined. The last time I saw the problem stated, the intended interpretation was crystal clear and exactly as you have said. >> Waving the magic wand, parameterizing by t and appealing to an intuitive >> notion of "if all steps of a process are defined it follows >> that the outcome is uniquely defined" is a poor substitute. That notion >> turns out to be false. >> > > I think you misunderstand me; I did not claim the above notion. > Certainly, it doesn't follow from what I've said that if we put ball 1 > in the vase at t = -1/n for odd n and remove ball 1 at t = -1/n for > even n, that therefore the outcome is uniquely defined at t>= 0. Nor > does it follow that every sequence V = {V_n} of subsets of naturals > therefore has a well-defined limit under pointwise convergence. Fair enough. I had that example in mind as well. >> >> In particular, arranging matters so that all the step times come before a >> >> particular finite time is irrelevant -- it's a trick designed to fool >> >> our intuitions into delusions of physicality and all the implicit >> >> assumptions that come with physicality. >> >> >> > >> > On the contrary, the red herring here is to assume that the problem is >> > of the form: let V = {V_n} be a sequence of subsets of N; what subset >> > of N which corresponds to the lim n->oo V_n? >> >> _The_ red herring? I don't think there is just one. >> > > Well, that is a matter of taste :). I certainly see people argue as if > the problem were solely one of an infinite sequence of sets of balls, > without noting that the problem already contains a notion of > convergence. Yes. I wouldn't go so far as to say that there is a single notion of convergence that is implicit in the problem, but there's certainly a requirement to have one. And there is an obvious topology to use to get one. >> > Now, there is an obvious notion of limit we can apply here
From: stephen on 13 Oct 2006 11:40
Dik T. Winter <Dik.Winter(a)cwi.nl> wrote: > In article <a1364$452f53c9$82a1e228$2828(a)news2.tudelft.nl> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > ... > > I don't know about Mueckenheim's opinion in these matters, but an old > > Greek philosopher - Heraclitos - has said: "panta rei kai ouden menei" > > (everything flows and nothing remains the same). And I agree with that. > Pray warn me when 2 has changed sufficiently to be the square of a rational. > I would not like to miss that moment. The day the circle is squared cannot be to far behind. Stephen |