Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: David Marcus on 9 Dec 2006 17:30 Tony Orlow wrote: > Is it sufficient to show that there are conclusions derived from > application of set theory that may not be mathematically correct in all > senses? No. You have to present mathematics that can't be formalized in ZFC. > 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? No. You have to present mathematics that can't be formalized in ZFC. Simply present your "other mathematical methods". Then we can see if they really need techniques that can't be modelled in ZFC. > 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. -- David Marcus
From: David Marcus on 9 Dec 2006 17:33 Han.deBruijn(a)DTO.TUDelft.NL wrote: > It's a good thing that you mention Church. He and others have proposed > quite another paradigm than set theory. I would like to call it LABOUR. > What do you think about the following: anything in mathematics can be > founded on labour ! To say it otherwise: all mathematics is: functions. Duh! Anyone could have told you that functions are a plausible foundation to build mathematics on. However, I'm not sure if this has been worked out in detail. In truth, most working mathematicians don't really care what foundation is used. > Ah, and don't say now that functions are sets. Church et al. have been > working the other way around, with sensible results, such as functional > programming (and languages, like good old LISP). -- David Marcus
From: David Marcus on 9 Dec 2006 17:35 Han.deBruijn(a)DTO.TUDelft.NL wrote: > 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. The usual model for time is the real numbers. And the real numbers can certainly be modelled in ZFC. I can't believe I have to say things that are so obvious. I mean, mechanics is always given as an example of the use of calculus, and mechanics is all about time. Then there is the mathematics behind special and general relativity. -- David Marcus
From: David Marcus on 9 Dec 2006 17:36 Virgil wrote: > 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. That's possible. -- David Marcus
From: Bob Kolker on 9 Dec 2006 17:36
David Marcus wrote: > > Fine. Give some evidence that it "narrows your thinking". I.e., present > some mathematics that can't be done using ZFC as a foundation. What about category in its full glory? Bob Kolker |