An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: David Marcus on 18 Oct 2006 18:00 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> MoeBlee wrote: > >>> P.S. > >>> > >>> I lost the context, but somewhere you (Orlow) posted: > >>> > >>> "ZF and NBG don't handle sequences or their sums, but only unordered > >>> sets" > >>> > >>> Z set theory defines and proves theorems about ordered tuples, finite > >>> and infinite sequences, and infinite summations and infinite products > >>> and many other things like that. > >>> > >>> MoeBlee > >>> > >> Huh! But I thought sets were unordered. > > > > Without the axiom of choice, we prove the existence of certain > > orderings on certain sets. With the axiom of choice, we prove that > > every set has a well ordering. > > > >> If the theory of infinite series > >> is derived from set theory, > > > > I don't say that historically it is derived from set theory. But > > infinite series, such as one finds in ordinary calculus and real > > analysis are definable in set theory and the theorems about them can be > > proven from the axioms of set theory. > > Theorems like, an infinite series with alternating positive and negative > terms can be validly rearranged to have a sum as large or small as one > likes? :) Sorry, I don't buy it. Which books have you read? -- David Marcus
From: David Marcus on 18 Oct 2006 18:04 Tony Orlow wrote: > Your examples of the circle and rectangle are good. Neither has a height > outside of its x range. The height of the circle is 0 at x=-1 and x=1, > because the circle actually exists there. To ask about its height at x=9 > is like asking how the air quality was on the 85th floor of the World > Trade Center yesterday. Similarly, it makes little sense to ask what > happens at noon. There is no vase at noon. Do you really mean to say that there is no vase at noon or do you mean to say that the vase is not empty at noon? -- David Marcus
From: David Marcus on 18 Oct 2006 18:08 Tony Orlow wrote: > You have agreed with everything so far. At every point before noon balls > remain. You claim nothing changes at noon. Is there something between > noon and "before noon", when those balls disappeared? If not, then they > must still be in there. I thought you just said that the vase doesn't exist at noon. If the vase doesn't exist, how can the balls be in it? -- David Marcus
From: David Marcus on 18 Oct 2006 18:09 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
From: David Marcus on 18 Oct 2006 18:12
Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> Virgil wrote: > >>> In article <4533d315(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> Then let us put all the balls in at once before the first is removed and > >>>>> then remove them according to the original time schedule. > >>>> Great! You changed the problem and got a different conclusion. How > >>>> very....like you. > >>> Does TO claim that putting balls in earlier but taking them out as in > >>> the original will result in fewer balls at the end? > >> If the two are separate events, sure. > > > > Not sure what you mean by "separate events". Suppose we put all the > > balls in at one minute before noon and take them out according to the > > original schedule. How many balls are in the vase at noon? > > > empty. Suppose we put ball n in at 1/n before noon and remove it at 1/(n+1) before noon. How many balls in the vase at noon? -- David Marcus |