Cantor Confusion
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From: David Marcus on 10 Dec 2006 00:52 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> Now, sequences may be said to derive from ordered sets, but sets are > >>>> said to be determined solely by membership, with order unimportant. So, > >>>> the notion of a sequence derives really from an inductive definition > >>>> such as Peano's, and not from the one primitive in set theory, > >>>> membership, alone. The notion of order is not captured by "is an element > >>>> of". Do you disagree? > >>> Of course I don't agree. You seem to be saying that infinite sequences > >>> can't be handled in ZFC. Since ZFC has no trouble modeling the natural > >>> numbers and defining functions, it clearly has no trouble acting as a > >>> foundation for all of calculus and analysis. > >> Is there not a single primitive in set theory, namely, e (element of)? > > > > Sure. But that just says that there is only one relation that is built > > into the language of ZFC. We are perfectly free to define new stuff, > > just as we do in any math class or book. > > > >> How is order derived from that, > > > > In the usual way. If we model the natural numbers as 0 = {}, 1 = {0}, 2 > > = {0,1}, then we can define n < m to mean n in m. We then define Z, Q, R > > in the usual way from N and define addition, multiplication, and order > > for all of them. Haven you seen the constructions? > > > > Ugh, yes, I guess I have. The von Neumann ordinals appear to be the > vehicle connecting set membership and order this way. Okay. I don't like > it, but it works in its way. It is just supposed to work. No one is saying zero really is the empty set (whatever "really is" means). > It seems like it would be better to have > another primitive, such as Peano's successor, than to use this strange > definition of the naturals, but I'll have to think about that. The idea is to be parsimonious. Since you can define the relation you want, there is no reason to be redundant by including another primitive. Doing so just makes things more complicated without any benefit. > >> and why is it not applicable in the case > >> of the staircase? > >> > >> For any given n, the number of steps, the staircase is defined as the > >> sequence of segment offset pairs: > >> (x=1->n: {(0,1/n),(1/n,0)} > > > > What do you mean "segment offset"? If n = 2, then you wrote something > > like > > > > (0,1),(1,0),(0,0.5),(0.5,0) > > You are reading it as if it said (x=1->n: {(0,1/x),(1/x,0)}, but 1/n is > constant for each iterated value of x. It would be > (0,0.5),(0.5,0),(0,0.5),(0.5,0). That is, up 1/2, right 1/2, up 1/2, > right 1/2. Oh. Simpler to just write the coordinates of the points as functions of x. > In other words, the pair of numbers denote the x and y > offsets from the start of the segment to its end. > > > Do you mean the staircase is the path connecting the points > > > > (0,0),(0,0.5),(0.5,0.5),(1,0.5),(1,1) > > > > ? > > Yes, that's what I meant, and that's what I said. We start at the > origin, (0,0). Then we add 1/n to the y value to get the next point, 1/n > to the x value to get the next, and repeat that n times, until we get to > (1,1). See? Each segment in the curve is defined by a single pair, the > starting point being already defined as the last ending point, and the > overall starting point being the origin. > > >> The diagonal may be divided into corresponding pairs of segments with > >> the same overall segment offset: > >> (x=1->n: {(sqrt(2)/2n,sqrt(2)/2n),(sqrt(2)/2n,sqrt(2)/2n)} > >> > >> The segments in the first are always vertical or horizontal, while those > >> in the second are all diagonal. The point set interpretation does not > >> catch this. Why? > > > > What "point set interpretation"? What doesn't it catch? > > I refer to the point set interpretation of the limit of the staircase, > put forth some time back by Chas as a counterexample to infinite-case > inductive proof. By his argument, since the points in the staircase > become arbitrarily close to the corresponding points in the diagonal as > n grows without bound, the staircase "in the limit" can be considered to > be the same object as the diagonal line. His conclusion is that, since > one can prove inductively that the length of the staircase is 2 for > every value of n, that proof only applies in the finite case, since his > version of the infinite case obviously has a length of sqrt(2). My > response to that was that the staircase in the limit is clearly a > different object than the diagonal line, preserving its right angles on > the infinitesimal scale, and therefore not straight, but a kind of > fractal object. The fact that the segments of the diagonal in the limit > are always at a 45 degree angle to the diagonal accounts for the > discrepancy in measure of sqrt(2), the inverse of the cosine of the > angle. In other words, approximating the diagonal using the staircase > segments cannot provide accurate measure, because the segments are not > parallel to the object they are approximating. This doesn't have anything to do with whether set theory can be used as a foundation for mathematics. You need to give a precise definition of what sort of limit you are doing that would allow the staircase to approach your "kind of fractal object". If you want such a thing, you have to invent it. Developing new mathematics isn't easy. > So, the point set approach has that the same object has two different > measures, because it cannot distinguish between two objects which are > locationally the same, and directionally different. Then come up with an approach that does what you want. By "approach", I mean definitions (of objects, convergence, etc.) that let you prove the theorems you want. That's what everyone does. For example, Einstein came up with Brownian motion, but it wasn't clear how to mathematically model it. Weiner figured out how. > I'm sure that cleared things up for you, eh? Pretty much. > By the way, the only other counterexample to infinite-case induction > suggested made obvious use of a discontinuity based on a buried > difference with a limit of 0 in the infinite case, and thus violated the > rules as I put forth. So, if you happen to have a good counterexample to > infinite-case induction that doesn't rely on such differences, I'd be > happy to decompose them for you. :) -- David Marcus
From: Han.deBruijn on 10 Dec 2006 08:47 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. > > Absolute rigour is a phantom, even in mathematics. > > Absolute lack of rigor is chaos, especially in mathematics. Absolute lack of rigor is chaos. But lack of absolute rigor is nice. Han de Bruijn
From: Tony Orlow on 10 Dec 2006 08:57 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> Now, sequences may be said to derive from ordered sets, but sets are >>>>>> said to be determined solely by membership, with order unimportant. So, >>>>>> the notion of a sequence derives really from an inductive definition >>>>>> such as Peano's, and not from the one primitive in set theory, >>>>>> membership, alone. The notion of order is not captured by "is an element >>>>>> of". Do you disagree? >>>>> Of course I don't agree. You seem to be saying that infinite sequences >>>>> can't be handled in ZFC. Since ZFC has no trouble modeling the natural >>>>> numbers and defining functions, it clearly has no trouble acting as a >>>>> foundation for all of calculus and analysis. >>>> Is there not a single primitive in set theory, namely, e (element of)? >>> Sure. But that just says that there is only one relation that is built >>> into the language of ZFC. We are perfectly free to define new stuff, >>> just as we do in any math class or book. >>> >>>> How is order derived from that, >>> In the usual way. If we model the natural numbers as 0 = {}, 1 = {0}, 2 >>> = {0,1}, then we can define n < m to mean n in m. We then define Z, Q, R >>> in the usual way from N and define addition, multiplication, and order >>> for all of them. Haven you seen the constructions? >>> >> Ugh, yes, I guess I have. The von Neumann ordinals appear to be the >> vehicle connecting set membership and order this way. Okay. I don't like >> it, but it works in its way. > > It is just supposed to work. No one is saying zero really is the empty > set (whatever "really is" means). > Then no one is saying the von Neumann successor ordinals "really are" the naturals? Good. >> It seems like it would be better to have >> another primitive, such as Peano's successor, than to use this strange >> definition of the naturals, but I'll have to think about that. > > The idea is to be parsimonious. Since you can define the relation you > want, there is no reason to be redundant by including another primitive. > Doing so just makes things more complicated without any benefit. > Not when the alternative is a construction that establishes the equivalent of a new primitive through a dubious connection between succession and containment. What is more "parsimonious" in inventing some weird model of the naturals and declaring that it exists, rather than having succ() be a primitive relation? I don't see the advantage. >>>> and why is it not applicable in the case >>>> of the staircase? >>>> >>>> For any given n, the number of steps, the staircase is defined as the >>>> sequence of segment offset pairs: >>>> (x=1->n: {(0,1/n),(1/n,0)} >>> What do you mean "segment offset"? If n = 2, then you wrote something >>> like >>> >>> (0,1),(1,0),(0,0.5),(0.5,0) >> You are reading it as if it said (x=1->n: {(0,1/x),(1/x,0)}, but 1/n is >> constant for each iterated value of x. It would be >> (0,0.5),(0.5,0),(0,0.5),(0.5,0). That is, up 1/2, right 1/2, up 1/2, >> right 1/2. > > Oh. Simpler to just write the coordinates of the points as functions of > x. > No, that is no simpler, and does not capture the direction or magnitude of any segment in a single pair. That's the "point". The pairs can denote absolute x and y coordinates for a point set perspective, or changes in x and y coordinates that describe the segment in a pair, the starting point of each segment determined by all previous segments, for s segment sequence perspective. Understand? >> In other words, the pair of numbers denote the x and y >> offsets from the start of the segment to its end. >> >>> Do you mean the staircase is the path connecting the points >>> >>> (0,0),(0,0.5),(0.5,0.5),(1,0.5),(1,1) >>> >>> ? >> Yes, that's what I meant, and that's what I said. We start at the >> origin, (0,0). Then we add 1/n to the y value to get the next point, 1/n >> to the x value to get the next, and repeat that n times, until we get to >> (1,1). See? Each segment in the curve is defined by a single pair, the >> starting point being already defined as the last ending point, and the >> overall starting point being the origin. >> >>>> The diagonal may be divided into corresponding pairs of segments with >>>> the same overall segment offset: >>>> (x=1->n: {(sqrt(2)/2n,sqrt(2)/2n),(sqrt(2)/2n,sqrt(2)/2n)} >>>> >>>> The segments in the first are always vertical or horizontal, while those >>>> in the second are all diagonal. The point set interpretation does not >>>> catch this. Why? >>> What "point set interpretation"? What doesn't it catch? >> I refer to the point set interpretation of the limit of the staircase, >> put forth some time back by Chas as a counterexample to infinite-case >> inductive proof. By his argument, since the points in the staircase >> become arbitrarily close to the corresponding points in the diagonal as >> n grows without bound, the staircase "in the limit" can be considered to >> be the same object as the diagonal line. His conclusion is that, since >> one can prove inductively that the length of the staircase is 2 for >> every value of n, that proof only applies in the finite case, since his >> version of the infinite case obviously has a length of sqrt(2). My >> response to that was that the staircase in the limit is clearly a >> different object than the diagonal line, preserving its right angles on >> the infinitesimal scale, and therefore not straight, but a kind of >> fractal object. The fact that the segments of the diagonal in the limit >> are always at a 45 degree angle to the diagonal accounts for the >> discrepancy in measure of sqrt(2), the inverse of the cosine of the >> angle. In other words, approximating the diagonal using the staircase >> segments cannot provide accurate measure, because the segments are not >> parallel to the object they are approximating. > > This doesn't have anything to do with whether set theory can be used as > a foundation for mathematics. You need to give a precise definition of > what sort of limit you are doing that would allow the staircase to > approach your "kind of fractal object". If you want such a thing, you > have to invent it. Developing new mathematics isn't easy. > If the sequence consists of segments of the form (0,x) or (x,0), there is no segment which is diagonal in direction. If every segment in the sequence is of the form (x,x), there is no segment which is not. This information is not evident when the pairs describing the curve are locations, because locations don't have direction. So, my question remains. If this is a valid formulation of the two objects, and an explanation for Chas' counterexample to infinite-case induction, where does this fit with set theory? I don't think the von Neumann ordinals as a model of a sequence can suffice, since they only allow finite values until a leap is made to the limit ordinals, and continuity is violated. This is the problem I have with the vNO's, and a large part of my problem with transfinitology. The notion that a sequence must be "countable" simply is not correct in the bigger picture. >> So, the point set approach has that the same object has two different >> measures, because it cannot distinguish between two objects which are >> locationally the same, and directionally different. > > Then come up with an approach that does what you want. By "approach", I > mean definitions (of objects, convergence, etc.) that let you prove the > theorems you want. That's what everyone does. For example, Einstein came > up with Brownian motion, but it wasn't clear how to mathematically model > it. Weiner figured out how. > Well, I'm suggesting a definition of the curve as a sequence of pairs which denote xy offsets, rather than a set of pairs of xy coordinates. Is that not a concrete enough description of an "approach" to spark a new neuron in your head? It should be. If you think there is something wrong with it, please elucidate. >> I'm sure that cleared things up for you, eh? > > Pretty much. Did it? I rather thought you'd accuse me of being totally nonsensical, though I know I'm not. Nice surprise (unless you're being as sarcastic as I was). Tony PS - No good counterexamples to infinite-case induction? Too bad. :( > >> By the way, the only other counterexample to infinite-case induction >> suggested made obvious use of a discontinuity based on a buried >> difference with a limit of 0 in the infinite case, and thus violated the >> rules as I put forth. So, if you happen to have a good counterexample to >> infinite-case induction that doesn't rely on such differences, I'd be >> happy to decompose them for you. :) >
From: Han.deBruijn on 10 Dec 2006 09:03 David Marcus schreef: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > > Ah, now don't act as if you didn't have those heated debates with some > > manifest opponents of set theory, i.e. Wolfgang Mueckenheim. > > If you've been reading the threads, then you should know that WM has so > far failed to present any mathematics at all. All he does is present > incorrect arguments that he insists follow from the standard axioms and > then proclaim: "Behold, standard mathematics is inconsistent." Big deal. Nonsense. The arguments presented by WM are quite reasonable. > Anyone can prove 2 = 1, if they bend a rule here and there. > > > I'm not > > talking about any possibilities to replace set theory by look-alikes. > > > > I'm talking about rejecting any monolithic foundation for mathematics, > > any "foundation" that narrows down my freedom of thinking. I hate any > > form of NewSpeak, whether it is called Set Theory, Category Theory or > > Object Oriented Programming. I've seen too many of these. > > Fine. Give some evidence that it "narrows your thinking". I.e., present > some mathematics that can't be done using ZFC as a foundation. Í'm pretty sure you can twist and bend any thought in such a way that it fits into your set theoretical paradigm. And as soon it doesn't fit, you'd simply say that it is not mathematics. Wolfgang Mueckenheim has come up with several of such examples. You'd not be better off with me. See e.g. the "Probability XOR Calculus" thread, which was initiated by this author: http://groups.google.nl/group/sci.math/msg/b686cb8d04d44962 Han de Bruijn
From: Han.deBruijn on 10 Dec 2006 09:19
stephen(a)nomail.com schreef: [ ... Thank you, Stephen, for this quite detailed explanation ... ] But then: > This whole calculation, including time, can be modelled > in set theory. It can't. Don't we have a debate with Wolfgang Mueckenhein about sets that change in time? With a negative outcome? Let us say that set theory is half the truth. Within set theory, any function is a special relation between commodities and products, i.e. domain and range. But the production process itself involves _labour_, hence time, and this aspect is not covered by the static set theoretic framework, where functions are reduced to "mappings" between sets. Han de Bruijn |