Cantor Confusion
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From: Tony Orlow on 9 Dec 2006 16:04 David Marcus wrote: > Han.deBruijn(a)DTO.TUDelft.NL wrote: >> stephen(a)nomail.com schreef: >> >>> Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: >>>> stephen(a)nomail.com wrote: >>>>> Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: >>>>> >>>>>> stephen(a)nomail.com wrote: >>>>>>> But everything can be modelled as a set. >>>>>> Define "everything" and prove that claim. >>>>> By "everything", I meant everything mathematical. Of course that is not 100% precise. >>>>> And no, I cannot prove it. But so far all the various objects of mathematics can be >>>>> modelled using set theory. That is what is meant by set theory being a foundation >>>>> for mathematics. If someone were to invent something "mathematical" (whatever that may >>>>> mean exactly) that could not be described in terms of set theory, then set theory would >>>>> no longer serve as a foundation. But given that the basics such as the real numbers, >>>>> functions, limits, calculus, etc. all can be founded in set theory, it would have to >>>>> be something strange indeed. Not that there is anything wrong with strange, but you >>>>> probably would like it less than set theory. >>>> Correction. By "everything" you probably mean "everything according to >>>> nowadays mainstream mathematics", which _is_, of course, "mathematics", >>>> according to your probably rather limited view. But since you can not >>>> really prove anything of the kind, I will rest my case. >>> It's not much of a case. You have not presented any evidence that there exists >>> any sort of mathematics not describable by set theory. Until such evidence >>> exists, the hypothesis that mathematics can be modelled with set theory has >>> not been falsified. And don't bother presenting something that uses limits, >>> functions, etc. as all of those things can be modelled with set theory. >> Ah, now you are trying to put the burden on me. But that is false play, >> of course. You said something like "all heat is phlogiston". I do not >> have to argue that this is not so. The burden remains yours. I didn't >> even say that set theory is useless within mathematics. I've only said >> that there's more to mathematics than set theory. > > All the mathematics I've ever seen in a math class or read in a math > book or journal or done myself can be done in ZFC. Admittedly, there are > large areas of mathematics that I know little or nothing about. Still, > that would seem to be quite a bit of evidence right there for the > statement that ZFC can be used as a foundation for mathematics. So, the > burden is now on you to show some mathematics that can't be done in ZFC. > <snip> Hi David - Is it sufficient to show that there are conclusions derived from application of set theory that may not be mathematically correct in all senses? If a conclusion based on premises of set theory does not match the conclusion based on other mathematical methods, then is there not a contradiction between the premises, and therefore premises which are not subsumed under set theory? 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. 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? Tony
From: Han.deBruijn on 9 Dec 2006 16:13 Virgil schreef: > In article <bed62$45797cde$82a1e228$28318(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > stephen(a)nomail.com wrote: > > > > > But everything can be modelled as a set. > > > > Define "everything" and prove that claim. > > > > Han de Bruijn > > Within ZF or ZFC everything that can be modeled must be modeled as a set > as that is all one has to model with. Sure. And now I have some threads about Chebyshev Polynomials with some *real* puzzles in them and nobody is responding. Because set theory is so powerful ? Let me finally experience any of this power, and maybe I will start believing you. Set Theory gas given me little that I didn't already know. And it has seldom helped me in problem solving. Han de Bruijn
From: Han.deBruijn on 9 Dec 2006 16:34 MoeBlee schreef: > Han de Bruijn wrote: > > 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. > > What pure mathematical calculation cannot be represented as a proof in > set theory? (And a proof, being a sequence of formulas, is, in Z set > theory as a meta-theory, a set.) Any calculation runs in TIME. And time is not a set. Han de Bruijn
From: Han.deBruijn on 9 Dec 2006 16:45 David Marcus schreef: > do just about anything mathematical and all we need are sets. Wrong. All you need is LOVE. But your sets are bound to kill my love for mathematics. Han de Bruijn
From: Virgil on 9 Dec 2006 16:47
In article <MPG.1fe44d0a78fa50dc989a24(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Virgil wrote: > > In article <MPG.1fe436bb3bd6003c989a18(a)news.rcn.com>, > > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > > > > Eckard Blumschein wrote: > > > > On 12/8/2006 1:05 AM, David Marcus wrote: > > > > > Eckard Blumschein wrote: > > > > >> On 12/5/2006 9:23 PM, Virgil wrote: > > > > >> > > > > >> >> Do not confuse Cantor's virtue of belief in god given sets with > > > > >> >> my > > > > >> >> power > > > > >> >> of abstraction. > > > > >> > > > > > >> > Cantor's religious beliefs are as irrelevant as EB's beliefs in > > > > >> > his > > > > >> > own > > > > >> > infallibility. > > > > >> > > > > >> I am not infallible. Show me my errors, and I will express my > > > > >> gratitude. > > > > > > > > > > Show you your errors or convince you that they really are errors? The > > > > > former is simple, but the latter appears to be impossible. We can't > > > > > force you to learn, if you don't wish to. > > > > > > > > I can force you to either refute e.g. my hint that Cantor's definition > > > > of a set has been declared untenable or tacitly accept this fact. > > > > > > Please restate the definition that you say is untenable. Let's take a > > > look. > > > > And tell us by whom it has been declared untenable, and why we should > > take his word for anything. > > I fear that if we ask for too many things at once, we are unlikely to > get any of them. Although admittedly, asking for one thing at a time > hasn't generally produced much. perhaps if we ask EB for enough things we will get something, rather than EB's usual nothings. s |