An uncountable countable set
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From: Ross A. Finlayson on 18 Oct 2006 23:46 David Marcus wrote: > Ross A. Finlayson wrote: > > Also, for each later time t less than noon, there are more balls in the > > vase than at time t, because for each difference there are ten balls > > added for each removed. > > > > So, how does the ball remove itself? > > > > Consider the ball and vase as a related rates tank problem. You pour > > in a dram of this syrupy liquid, and it goes to the bottom of the tank, > > where there's a hole in the bucket, Elvira, and the liquid drains out. > > That would seem similar to your problem here with the balls and vase, > > no? It even offers a mechanism, where you need to expand on some of > > the explanation there of the mechanics of the balls and vase. > > > > Your non-explanation is not a constraint, it's handwaving. > > > > So, the tank obviously fills. > > How would you translate the following problem into English ("balls", > "vase", "time")? > > Problem: For n = 1,2,..., define > > A_n = 12 - 1 / 2^(floor((n-1)/10)), > R_n = 12 - 1 / 2^(n-1). > > For n = 1,2,..., define a function B_n by > > B_n(t) = 1 if A_n < t < R_n, > 0 if t < A_n or t > R_n, > undefined if t = A_n or t = R_n. > > Let V(t) = sum{n=1}^infty B_n(t). What is V(12)? > > -- > David Marcus For n > 1, A_n < R_n < 12 for all n, so B_n(12) = 0 for all n, so, the sum of the B_n's is zero. As Cauchy sequences: R_n is 12, A_n is 12, and B_n(12) is 0. Compute the difference of R_n - A_n D_n = 1 / 2^(floor((n-1)/10)) - 1 / 2^(n-1) = ( 0, 1/2, 3/4, 7/8, 15/16, 31/32, 63/64, 127/128, 255/256, 255/512, 511/1024, 1023/2048, ... ) Notice it's not monotonic, in that not D_n+1 < D_n. So, the standard limit doesn't exist, where people used to say monotonicity was a requirement. I'd agree the limit exists and is zero, because 12-A_n goes to zero and 12-R_n < 12-A_n. The required vase is infinitely huge. In fact, it would have to be so huge, the balls are already in the vase, else there's not room for them. In the parallel to the related rates tank problem, almost all the syrup overflows and is lost, besides that as an incompressible fluid the pressure would shatter any finite tank. A rabbit that runs unimpeded at 10 m/s is bound to a 1 m/s turtle, with the turtle trodding the other way the rabbit does 9 m/s. If your vase is empty you have that the turtle always beats Achilles if it starts a fraction ahead. Please explain the mechanics of the vase. How do you retrieve the selected ball at time t? Find a real world situation, with photons and a black hole or something, where you might actually explain the mechanics of this completion of infinity and etc., else it's quite unrealistic. Ross
From: David Marcus on 18 Oct 2006 23:51 Han de Bruijn wrote: > David Marcus wrote: > > Han de Bruijn wrote: > >>No matter how much textbooks about set theory I read, it all remains > >>abacedabra for me. > > > > I just looked in two books on set theory on my shelf and both explicitly > > state that the set in the axiom of infinity need not be the natural > > numbers. The books are "Naive Set Theory" by Paul R. Halmos and "Set > > Theory, An Introduction to Independence Proofs" by Kenneth Kunen. > > Hmm, I have that book by Halmos myself ... Then look at the discussion right after the axiom of infinity on page 44. > But why are the finite ordinals not equivalent with the naturals (I mean > in mainstream mathematics)? First you said that the axiom of infinity is the construction of the ordinals. We said no. Then you said the axiom of infinity defines the finite ordinals. We said no. Now you say the finite ordinals are the natural numbers (actually you said "equivalent", but I'm not sure what that means). Third time's the charm, i.e., you are correct. A standard definition of the natural is as the finite ordinals. However, you seem to think that all three of your statements are the same. I don't understand this. > > Which math textbooks have you read? Have you worked the problems in the > > ones you've read? Taken any math courses at a university? > > I'm a (theoretical) physicist by education. Does that mean you have taken graduate courses in math (e.g., set theory) or you haven't? -- David Marcus
From: David Marcus on 19 Oct 2006 01:05 Ross A. Finlayson wrote: > David Marcus wrote: > > Ross A. Finlayson wrote: > > > Also, for each later time t less than noon, there are more balls in the > > > vase than at time t, because for each difference there are ten balls > > > added for each removed. > > > > > > So, how does the ball remove itself? > > > > > > Consider the ball and vase as a related rates tank problem. You pour > > > in a dram of this syrupy liquid, and it goes to the bottom of the tank, > > > where there's a hole in the bucket, Elvira, and the liquid drains out. > > > That would seem similar to your problem here with the balls and vase, > > > no? It even offers a mechanism, where you need to expand on some of > > > the explanation there of the mechanics of the balls and vase. > > > > > > Your non-explanation is not a constraint, it's handwaving. > > > > > > So, the tank obviously fills. > > > > How would you translate the following problem into English ("balls", > > "vase", "time")? > > > > Problem: For n = 1,2,..., define > > > > A_n = 12 - 1 / 2^(floor((n-1)/10)), > > R_n = 12 - 1 / 2^(n-1). > > > > For n = 1,2,..., define a function B_n by > > > > B_n(t) = 1 if A_n < t < R_n, > > 0 if t < A_n or t > R_n, > > undefined if t = A_n or t = R_n. > > > > Let V(t) = sum{n=1}^infty B_n(t). What is V(12)? > > > For n > 1, A_n < R_n < 12 for all n, so B_n(12) = 0 for all n, so, the > sum of the B_n's is zero. Thank you. But, I didn't ask for your solution to the problem. I asked how you would translate the problem into English, i.e., using the words "balls", "vase", and "time'. So, please translate the problem into English. > As Cauchy sequences: R_n is 12, A_n is 12, and B_n(12) is 0. > > Compute the difference of R_n - A_n > D_n = 1 / 2^(floor((n-1)/10)) - 1 / 2^(n-1) > > = ( 0, 1/2, 3/4, 7/8, 15/16, 31/32, 63/64, 127/128, 255/256, > 255/512, 511/1024, 1023/2048, ... ) > > Notice it's not monotonic, in that not D_n+1 < D_n. So, the standard > limit doesn't exist, where people used to say monotonicity was a > requirement. I'd agree the limit exists and is zero, because 12-A_n > goes to zero and 12-R_n < 12-A_n. > > The required vase is infinitely huge. In fact, it would have to be so > huge, the balls are already in the vase, else there's not room for > them. > > In the parallel to the related rates tank problem, almost all the syrup > overflows and is lost, besides that as an incompressible fluid the > pressure would shatter any finite tank. > > A rabbit that runs unimpeded at 10 m/s is bound to a 1 m/s turtle, with > the turtle trodding the other way the rabbit does 9 m/s. If your vase > is empty you have that the turtle always beats Achilles if it starts a > fraction ahead. > > Please explain the mechanics of the vase. How do you retrieve the > selected ball at time t? > > Find a real world situation, with photons and a black hole or > something, where you might actually explain the mechanics of this > completion of infinity and etc., else it's quite unrealistic. I agree it is not realistic. -- David Marcus
From: imaginatorium on 19 Oct 2006 02:56 David Marcus wrote: > imaginatorium(a)despammed.com wrote: > > Ross A. Finlayson wrote: > > > I actually recommend that you read the book "Counterexamples in Real > > > Analysis." That contains scores of what are called "counterexamples in > > > standard real analysis." > > > > Sorry, Ross, perhaps I misread. Um, well, on looking closely, perhaps I > > didn't. Here's what you refer to: > > > > "counterexamples to standard real analysis" > > > > I read that as meaning that there are counterexamples that show that > > standard real analysis is inconsistent, wrong, or otherwise defective. > > But the title of the book you mention is subtly different: > > > > "Counterexamples in Real Analysis" > > > > I went to Amazon.com, and found a book with that title, and looked in > > the front. I saw a list of "counterexamples", and noticed that one of > > them was (from memory) "There exists a set with an infimum greater than > > its supremum". Sounds very odd - intuitively, it seems obvious that, > > assuming the Latin words mean greatest lower bound and least upper > > bound, the glb must be less than the lub, *until* you think about > > proving it, and think about the definitions. Aha! Consider the reals in > > [0, 1] with the usual ordering, then these have a subset whose glb is 1 > > and whose lub is 0. (At least if I've guessed the standard definitions > > correctly.) Can you see what it is? > > "inf" is glb and "sup" is lub, as you surmised. According to Amazon, the > book says there exists a subset of R, not a subset of [0,1]. I wonder > how the book is defining inf and sup, since with the definitions I > usually see, inf A <= sup A, where A is any subset of R where the inf > and sup exist. Well, I'd forgotten exactly what it said by the time I wrote the above. I assume that if an upper bound is defined as: p is an upper bound of set S, iff for all s in S, p >= s And if P is the (non-empty) set { p : p is an ub } then the lub q is the smallest p. (Exercise: prove it exists) Considering subsets of [0, 1], if X is the empty set, then every element of [0, 1] is both ub and lb. So the glb is 1, and the lub is 0. I don't see how this can happen for the reals though. But perhaps if the set of upper bounds is empty, you define the lub as the formal symbol m, and similarly the glb of a set unbounded below as w, then the lub of the empty set is w, and the glb is m. (Question for Tony: Does this mean that w and m are real numbers?) In a sense, it seems to me, the core of the problem is here. Mathematics is all about being free to make your own definitions, so it's almost impossible to say of any sentence full of mathematical words, but taken in isolation, that it is wrong. People used to looking at different books, and seeing different treatments, try to interpret superficially surprising statements by guessing at the writer's own definition. The problem is that they when the writer is a crank, there _is_ no definition, only the words. Hmm, perhaps that's obvious. Brian Chandler http://imaginatorium.org
From: Han de Bruijn on 19 Oct 2006 03:21
MoeBlee wrote: > Han de Bruijn wrote: > >>But why are the finite ordinals not equivalent with the naturals (I mean >>in mainstream mathematics)? > > The set of finite ordinals IS the set of natural numbers. > > x is a natural number <-> x is a finite ordinal. > > Why don't you just read a textbook? It's all very confusing. Because there also "exist" infinite ordinals, they say. Han de Bruijn |