Cantor Confusion
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From: William Hughes on 12 Oct 2006 20:19 Tony Orlow wrote: > William Hughes wrote: > > Tony Orlow wrote: > >> William Hughes wrote: > >>> Tony Orlow wrote: > >>>> Dik T. Winter wrote: > >>>>> In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> "Albrecht" <albstorz(a)gmx.de> writes: > >>>>> > David Marcus schrieb: > >>>>> ... > >>>>> > > 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. > >>>>> > > >>>>> > What is the difference to the diagonal argument by Cantor? > >>>>> > >>>>> That a (to the right after a decimal point) infinite string of decimal > >>>>> digits defines a real number, but that a (to the left) infinite string > >>>>> of decimal digits does not define a natural number. > >>>> It defines something. What do you call that? If the value up to and > >>>> including every digit is finite, how can the string represetn anything > >>>> but a finite value? > >>>> > >>> Because there are two types or strings. Strings that end and strings > >>> that don't end. Only strings that end represent finite values. > >>> > >>> -William Hughes > >>> > >> And what about countably infinite strings which cannot achieve actually > >> infinite values? > > > > If the strings do not end they do not represent finite values. > > Please prove this in specific mathematical terms, without using the > circular argument involving the supposed infinity of omega. > > > > > You have a hole between what most people call finite and > > what you call infinite. > > Uh, no, you do. > > You argue that since the strings > > cannot represent infinite things they must represent finite > > things. This does not follow. > > So, they can represent a third kind of number, which is not finite, nor > infinite? Pray tell, what sort be's such a number? Who says they have to represent a number? You have said that the size of an unbounded set of integers that does not contain an infinite integer is not any kind of integer. > > You need at least three > > catagories, finite, unbounded and infinite. > > Well, I have been lambasted for calling the countably infinite "finite > but unbounded". Are those two mutualy exclusive? Is there an "unbounded" > which is not "infinite? In standard mathematics an unbounded set of integers must be infinite. However, you claim that an unbounded set of integers cannot be infinite. However this does not make an unbounded set of integers finite (in the sense that an unbounded set of integers does not have the properties that are usually associated with a finite set. E.g. an unbounded set of integers does not have a bound ( you can of course use the term finite to mean anything you want, just don't be surprised that using finite in a non-standard way leads to confusion)). > > Please answer carefully, as I would not want you to make any mistakes. > > > Countably infinite strings have unbounded length and > > do not represent finite values. > > > > *Because* they are unbounded? > Yes, according to you the length of a countably infinite string is not any type of integer. I suppose one could have a "finite value" which is not an integer, but you don't argue this. >Does "unbounded" MEAN *infinite*? > In standard mathematics yes. In TO land, no. > > [ You have four types of ordered sets of integers. > > > > -sets that end with a finite integer > > -sets that do not end but do not contain an infinite integer > > -sets that end with an infinite integer > > -sets that do not end and do contain an infinite integer > > > > Do you actually allow these for types of sets of quantities? > "You have four types of ordered sets of integers" means "TO has four types of ordered sets of integers". This is just simple observation. Examples (Let M be any TO infinite integer.) {1,2,3} the finite evens {1,2,3,...,M} {... M-3,M-2,M-1,M,M+1,M+2,M+3...} (i.e. the set of all integers that have a finite difference with M) > > You have never really come to terms with the second type of set. > > Those are the T-riffics and their family. > As I said, you have never really come to terms with the second type of set. > > (You have never really come to terms with the fourth type of set, > > but this type of set has rarely been discussed). > > Of course they have, but like unbounded sets of finites, they're rather > hard to measure, For "rather hard to measure" read "I can't make them work in my system at all so I prefer to ignore them" > especially given that one has to accept first of all > the notion of a specific infinite quantity. > Surely you are not claiming that the set {... M-3,M-2,M-1,M,M+1,M+2,M+3...} has infinite size? - William Hughes
From: MoeBlee on 12 Oct 2006 20:41 Han de Bruijn wrote: > Set Theory is simply not very useful. The main problem being that finite > sets in your axiom system are STATIC. They can not grow. Set theory provides for capturing the notion of mathematical growth. Sets don't grow, but growth is expressible in set theory. If there is a mathematical notion that set theory cannot express, then please say what it is. > (I wouldn't imagine the situation that a table > in a database would have to be redefined, every time when a new row has > to be inserted, updated or deleted ...) That's a question of naming and defining, not a question of a set being entirely extensional. This is not a problem for set theory. If you "add a row to a matrix", then it is a different matrix that "results". But what those matrices have in common is expressible in set theory, so we can capture the notion of a "changing matrix" even though the "change" is functional, not a changing object itself. MoeBlee
From: Mike Kelly on 12 Oct 2006 21:08 Tony Orlow wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Dik T. Winter wrote: > >>> In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> "Albrecht" <albstorz(a)gmx.de> writes: > >>> > David Marcus schrieb: > >>> ... > >>> > > 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. > >>> > > >>> > What is the difference to the diagonal argument by Cantor? > >>> > >>> That a (to the right after a decimal point) infinite string of decimal > >>> digits defines a real number, but that a (to the left) infinite string > >>> of decimal digits does not define a natural number. > >> It defines something. > > > > But not necessarily a number. > > > >> What do you call that? If the value up to and > >> including every digit is finite, how can the string represetn anything > >> but a finite value? > > > > Because representations of finite values end, and the string doesn't > > end, so it breaks the rules of "strings that represent finite values". > > > > - Randy > > > > Can you rightly call it an infinite value? I can't. It's unbounded like > the finites themselves, but not infinite, as long as all digit positions > are finite. Maybe it doesn't represent a value at all? -- mike.
From: cbrown on 12 Oct 2006 21:34 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). (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. > > 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. > >> 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. > > Now, there is an obvious notion of limit we can apply here (pointwise > > convergence); but as Dik asserted, in that case we need to /define/ > > that notion, independent of the given formulation of the problem. > > > > This opens the door to complaints such as "noon never arrives" (which > > is essentially the claim that there is no such thing defined in the > > problem as the limit of the sequence V); or that some other form of > > convergence should be used. > > > > For example, suppose the steps occured at t=1, t=2, etc. We still have > > the total ordering you describe below, but now one can argue that "when > > all steps are completed, the vase is empty" is a meaningless statement, > > because there is no such time "when" this state is achieved. > > My point is that the problem is equally meaningful or meaningless > whether you augment your t scale by contemplating t=oo or whether you > scale so that t=12 occurs after all relevant events. > Heavens no!! This is precisely t
From: cbrown on 12 Oct 2006 22:36
MoeBlee wrote: > Han de Bruijn wrote: > > Set Theory is simply not very useful. The main problem being that finite > > sets in your axiom system are STATIC. They can not grow. > > Set theory provides for capturing the notion of mathematical growth. > Sets don't grow, but growth is expressible in set theory. If there is a > mathematical notion that set theory cannot express, then please say > what it is. > > > (I wouldn't imagine the situation that a table > > in a database would have to be redefined, every time when a new row has > > to be inserted, updated or deleted ...) > > That's a question of naming and defining, not a question of a set being > entirely extensional. This is not a problem for set theory. If you "add > a row to a matrix", then it is a different matrix that "results". But > what those matrices have in common is expressible in set theory, so we > can capture the notion of a "changing matrix" even though the "change" > is functional, not a changing object itself. > This is a thing that I find hard to understand: those who self-identify with the label "anti-Cantorian" do not comprehend that the language of set theory is simply a very useful way of describing a particular mathematical state of affairs. Is there a way, in the language of set theory, to state "although there is no largest set which is finite, there exists no set which is infinite"? Of course! That's exactly what set theory /is for/. Cheers - Chas |