Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: cbrown on 9 Dec 2006 21:40 Han.deBruijn(a)DTO.TUDelft.NL wrote: <snip> > What do you think about the following: anything in mathematics can be > founded on labour ! To say it otherwise: all mathematics is: functions. Have you read any Category Theory? You might find it interesting. Equally possibly, you might find that it only compounds your confusion. Cheers - Chas
From: Tony Orlow on 9 Dec 2006 23:02 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 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. >> 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. 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. 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. I'm sure that cleared things up for you, eh? 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. :) Tony
From: stephen on 9 Dec 2006 23:17 Han.deBruijn(a)dto.tudelft.nl wrote: > David Marcus schreef: >> Han.deBruijn(a)DTO.TUDelft.NL wrote: >> > To those who are unable to see it, there is no evidence that there is >> > any sort of mathematics not describable by set theory. But this is a >> > vicious circle. Mainstream mathematics simply DOES NOT ALLOW any sort >> > of mathematics that violates set theory. >> >> Quite a silly thing to say. Do you have any evidence for such an absurd >> statement? [ ... snipped "counter evidence" with Category Theory ... ] > 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. 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. > Han de Bruijn You are the one who is trying to restrict peoples' freedoms. You cannot name a single piece of mathematics that is "forbidden" by set theory. You on the other hand want to abolish any mathematics that involves infinity, which is a severe restriction on others freedom of thinking. In short, you are a hypocrite. I would imagine your understanding of Object Oriented Programming is as woefully garbled as you understanding of set theory. Stephen
From: imaginatorium on 10 Dec 2006 00:04 David Marcus wrote: > Tony Orlow wrote: <snip> > > The infinite staircase comes to mind, where point set topology considers > > the limit of the staircase from (0,0) to (1,1), as the number of steps > > increases without bound, to be the same object as the diagonal line from > > (0,0) to (1,1), since the locations of the corresponding points become > > arbitrarily close. This produces a contradiction in measure, the object > > being of length 2 for all staircases, but of length sqrt(2) for the > > diagonal line. While the locations of the points in each set approach > > each other with no lower limit, the directions of the corresponding > > sub-segments of the two objects are always at a 45 degree angle to each > > other, producing the error of sqrt(2)/2, the cosine of that angle. So, > > what we have are a diagonal line of length sqrt(2) and a fractal "line" > > or curve of length 2. In other words, characterizing the objects as sets > > of points misses the distinction between the objects in terms of > > measure, whereas characterizing them as sequences of segments preserves > > the distinction in terms of direction and overall length. > > Present your mathematics by itself. Then we can see if you are using > something other than what is in ZFC. > > > 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. But ZFC does have considerable difficulty dealing with infinite sequences, when the Orlovian axioms are added in. I _think_ that Tony's "infinite induction" thing means for a start that if a sequence of elements has a particular property ("staircase length is 2") then the limit of the sequence must have the same property. There doesn't seem to be a definition of the Orlovian limit, except that in any particular case Tony will construct an ad hoc story to make something having the property being talked about. Thus the Tlimit of the staircase sequence is a [search the archive for TO's words] "sort of infinitesimal staircase-thingy of length 2". I don't think ZFC will handle Orlovian "positive infinite quantities" too well, either. Tony gets to infinite values by simply advancing along the real line for, um, an infinite distance, through the tunnel of love (where it's too dark to see properly). Despite the fact that any finite quantity (integer) can be represented as a "finite length" two-ended string of digits, and the fact that each new integer is formed by adding one, the naive expectation that this would enable a proof by induction that these "infinite quantities" were also simply finite quantities, it doesn't work like this, because, um, there's a principle of somethingorother that excludes this. I wonder if in fact just as electromagnetic radiation is mediated by photons, induction is mediated by inductons, and the tunnel of love just happens to block the passage of inductons. Perhaps. Brian Chandler http://imaginatorium.org
From: stephen on 10 Dec 2006 00:22
David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > stephen(a)nomail.com wrote: >> Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: >> > Lester Zick wrote: >> >> On Thu, 7 Dec 2006 03:00:21 -0500, David Marcus >> >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >> >> >>>If you use ZFC (or something similar) as your foundation for >> >>>mathematics, then everything is a set. Of course, while solid >> >>>foundations are good to have, if you are living on an upper floor, you >> >>>may prefer to ignore what is going on in the basement. >> >> >> >> So you're saying that set "theory" is all of mathematics? Of course >> >> since what you say isn't necessarily true that's not exactly a ringing >> >> endorsement of set "theory". >> >> > It's quite simple. Set Theory can not be the foundation for mathematics, >> > because NOT EVERYTHING IS A SET. E.g. a calculation is mathematics, but >> > it's not a set. Set theory may be of limited use, but it's supremacy is >> > complete nonsense, and will be overruled in time. >> >> But everything can be modelled as a set. You simply do not understand >> what "foundation" means in this context. Any calculation can >> be rewritten as a set theory problem. It would be long, cumbersome, >> and impractical, but it could be done. Just as an computer program >> can be transformed into a Turing machine. > Yes, but I think it is a bit simpler than that. The study of algorithms > involves the study of certain functions on certain number systems. Such > functions and numbers can be handled in ZFC. That's the real point of > the construction of the natural numbers from sets: to show that set > theory can be used as the foundation for arithmetic and hence analysis, > etc. However, if you are doing numerical analysis or calculus, you think > of N, Z, Q, R, C, etc. as primitive objects. You don't care that they > can be modelled as sets. But writing down all the details using nothing but sets would be cumbersome. Just saying that numbers can be represented as sets is not the same as actually demonstrating how a calculation would be represented using sets and nothing but sets. As I said, it is like writing a Turing machine that computes a Fourier transform. Anyone who understands the theory of computation knows that it can be done, but nobody is likely to do it, especially just to satisfy who has no understanding of the theory of computation. Stephen |