Cantor Confusion
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From: David Marcus on 9 Dec 2006 17:43 Bob Kolker wrote: > 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? That is a reasonable example. However, I doubt that Han makes much use of it. Also, I don't think there is anything taught in undergraduate mathematics that needs more than ZFC. -- David Marcus
From: Virgil on 9 Dec 2006 17:47 Han.deBruijn(a)DTO.TUDelft.NL wrote: > 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. "Object Oriented Programming"? HdB is a bit off base here, as OOP is no part of mathematics. Perhaps if HdB had some idea of what he was talking about...
From: Bob Kolker on 9 Dec 2006 17:58 Virgil wrote: > > > "Object Oriented Programming"? > > HdB is a bit off base here, as OOP is no part of mathematics. Not quite. About ten years ago I ran into a categorical theoretic formulation of objects and classes (in the programming sense). I wish I could remember the reference. Bob Kolker
From: Tony Orlow on 9 Dec 2006 19:52 David Marcus wrote: > 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. > Is there not a single primitive in set theory, namely, e (element of)? How is order derived from 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)} 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?
From: David Marcus on 9 Dec 2006 20:51
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? > 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) 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) ? > 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? -- David Marcus |