Cantor Confusion
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From: David Marcus on 10 Oct 2006 21:00 Albrecht wrote: > Arturo Magidin wrote: > > In article <1159410937.013643.192240(a)h48g2000cwc.googlegroups.com>, > > <the_wign(a)yahoo.com> wrote: > > >Cantor's proof is one of the most popular topics on this NG. It > > >seems that people are confused or uncomfortable with it, so > > >I've tried to summarize it to the simplest terms: > > > > > >1. Assume there is a list containing all the reals. > > >2. Show that a real can be defined/constructed from that list. > > >3. Show why the real from step 2 is not on the list. > > >4. Conclude that the premise is wrong because of the contradiction. > > > > This is hardly the simplest terms. Much simpler is to do a ->direct<- > > proof instead of a proof by contradiction. > > > > 1. Take ANY list of real numbers. > > 2. Show that a real can be defined/constructed from that list. > > 3. Show that the real from step 2 is not on the list. > > 4. Conclude that no list can contain all reals. > > > > This summarization of the diagonal argument of Cantor seems to be > accepted by the most people in sci.math. > I like to examine the idea of Russell Easterly - building a kind of > diagonal number on lists of natural numbers - in respect to this view. > First I review his idea: > > Let's have an arbitrary list of natural numbers: > > 1: a > 2: b > 3: c > ... > > with the numbers of digits of the numbers in the list: > > 1: m > 2: n > 3: o > ... > > Now we build the "diagonal number" d of the list as follows: > > We have a look on the first number of the list a which is build out of > m digits. We build a number with m+1 digits with the cipher 1. This > number is truely greater than a and therefore different from a. > It's the first approach to d. (E.g. a = 765 -> m = 3 -> d = 1111) > Now we have a look on the second entry of the list. The number b with n > digits. If n <= m we let d unchanged. If n > m we build the new number > d with n+1 digits, again with the cipher 1. > In this way we go through the list. > > This construction builds up a number d which is different from any > number of the list. > > Now we want to test this sentences: > > 1. Take ANY list of natural numbers. > 2. Show that a natural number can be defined/constructed from that > list. > 3. Show that the natural number from step 2 is not on the list. > 4. Conclude that no list can contain all natural numbers. > > 1.: Any lists are any finite and infinite lists of natural numbers > (Axiom of infinity). > 2.: The number d has a finite, integer difference to any number of the > list by construction. A number which could be written as the sum of two > natural numbers is a natural number too. I don't follow. How do you know that the procedure that you gave actually "defines/constructs" a natural number d? It seems that you keep adding more and more digits to the number that you are constructing. > 3.: The number d is different to any number of the list by > construction. > 4.: The natural numbers are nondenumerable. > > What's wrong with Russell's argument but right with Cantor's? -- David Marcus
From: cbrown on 10 Oct 2006 21:06 David Marcus wrote: > David R Tribble wrote: > > Dik T. Winter wrote: > > >> 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. > > > > > > > Virgil wrote: > > >> Can you suggest one? One that does not ignore the numbering on the balls > > >> as some others have tried to do. > > > > > > > David Marcus wrote: > > > Models that ignore the number of the balls are certainly models. Whether > > > they are reasonable translations of the original problem into > > > Mathematics is a separate question. > > > > I tried modeling the problem with sets: > > B_0 = { } > > B_1 = B_0 U {1,2,3,...,10} \ {1} > > ... > > B_n = B_(n-1) U {10(n-1)+1,10(n-1)+2,...,10(n-1)+10} \ {n} > > ... > > > > The resulting "balls in the vase at noon" would then seem to be B_w. > > Which is problematic, since there is no ball labeled "w". So this is > > probably a bad model of the problem. (I don't claim to be an expert.) > > I think you are still using the ball labeling. So, even if your model > gave an answer for noon, I don't think it is what Virgil asked for. > > You could define B_w = {x | exists m such that for all n > m, x in B_n}. > Another possibility is B_w = {x | for all m, there is an n > m such that > x in B_n}. > In both your and David Tribble's examples, I don't see any reference to what time t step n occurs. This information is required to determine "the number of balls at time t" for /any/ time t (not just noon). Cheers - Chas
From: Dik T. Winter on 10 Oct 2006 21:41 In article <1160496764.343276.115430(a)k70g2000cwa.googlegroups.com> Han.deBruijn(a)DTO.TUDelft.NL writes: > David R Tribble schreef: > > Mueckenheim wrote: > > >> But you cannot derive that the vase is not empty at noon from the > > >> observation that its contents cannot decrease? > > > > Han de Bruijn wrote: > > > A picture says more than a thousand words. [Doesn't] it? > > > > > > http://hdebruijn.soo.dto.tudelft.nl/jaar2006/ballen.jpg > > > > I notice that there is no Y point at the rightmost X at "noon". > > True. That symbolizes the fact that there is no noon. Ah, a reincarnation of Zeno. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: cbrown on 10 Oct 2006 21:54 William Hughes wrote: > imaginatorium(a)despammed.com wrote: > > Dik T. Winter wrote: > > > In article <1160377400.288823.275240(a)c28g2000cwb.googlegroups.com> cbrown(a)cbrownsystems.com 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? > > > > > > > > I disagree. > > > > > > Do you? > > > > > > > This is no more difficult than asking "how do you *define* the number > > > > of balls at pi/10 seconds before midnight?" > > > > > > And you start giving a definition. I do not say that it is *difficult* to > > > define. But in my opinion more than one definition is possible. > > > > I agree with Chas - this simply isn't anything to be _defined_. Of > > course the problem as stated is in real-worldy-looking terms, as though > > this were plausibly an experiment one could do (can you say "intuition > > pump"), and it is necessary to _define_ how the statement is to be > > interpreted mathematically. This seems to be fairly simple ... > > I straddle the fence a bit. I agree with Dik Winter that a defintion > of the set of balls in the vase at noon is needed, however, > I agree with you that there is one very obvious definition of > this set (the balls that are added before noon and not > removed before noon), > nor can I see any other sensible definitions (clearly one > can create arbitrary definitions, e.g. the set of balls at noon > consists of the ball 47, but these are not very interesting). > > The problem is that the question asks not > > i. What is the set of balls in the vase at noon. > > but > > ii. How many balls are in the vase at noon. > > One obvious approach to answering ii is > > -define the set of balls in the vase at noon > -determine what this set is > -determine the cardinality of the set > > However, there is an alternate approach. Define > the number of balls in the vase at noon directly. Here > we have two intuitive approaches > > a: the number of balls in the vase at noon > is the cardinality of the set of balls in the vase at > noon. > > b: the number of balls in the vase at noon is the limit > of the number of balls in the vase as t approaches noon > But this is a circular definition of "the number of balls in the vase" unless we can agree that before noon, "the number of balls in the vase at time t" is the cardinality of the set of balls in the vase at time t. Once we have agreed to that definition, the only reason to change it that I can see is if we cannot determine what the cardinality of set of balls in the vase at t = 0. > a and b are of course contradictory if we use the "natural" definiton > of the set of balls in the vase at noon. How could one possibly deduce that the problem statement is not compatible with the following two statements: "If the number of balls in the vase at time t is not 0, then there is a ball in the vase." "If there is a ball in the vase, it has a natural number on it." > Those who want > to insist on definiton b have three choices > > I Infinite sets are contradictory i.e., "Math is hard!" :) > (however this contradiction > depends on b being true, and there is no proof within > set theory that b is true, so this "contradiction" is based > on intuition) > More exactly, the assumption that the number of balls in the vase at noon is the limit of the sequence of numbers of balls at each of the steps contradicts one or both of the two statements above. > II The number of balls in the vase at noon has nothing > to do with the cardinality of the set of balls in the vase at > noon (this option is not attractive, and no one chooses it) > Because it would be silly. > III Use some other definition for the set of balls in the > vase than the "natural" definition. (Although at times > the necessity of doing this is admitted, the intuitive appeal > of the "natural" definiton is so strong that it has not > been done. At best a set of magic balls, consisting > of balls labelled with "undeterminable integers", or > "infinite integers" is posited. The properties of these > balls are not specified, indeed it may be said that the > properties cannot be specified, hence it cannot be known > that these balls are not in the vase. Since some balls must > be in the vase, these balls must be in the vase > (why these balls and not some other balls such as the > "hyper-indeterminable" balls is not made clear).) > This is the denial of the statement "if a ball is in the vase, it has a natural number on it." I find it hard to justify as a problem arising from something not being "well-defined" in the original problem; in fact it seems quite contradictory to the observation that at no step in the process do we place anything in the vase that is not a ball with a natural number on it. Cheers - Chas
From: Dik T. Winter on 10 Oct 2006 22:00
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). You state (with your model) that there are no balls at noon. I state (with another model) that there is one ball at noon. What you want is to disprove the alternative model, but based on the question alone, that alternative model can not be proven false. You need quite a few axioms to *prove* it false. And, so, the question as is is understated, because not enough context is given. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |