An uncountable countable set
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From: cbrown on 23 Oct 2006 17:27 David Marcus wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > > > > Well, allow me to repeat them here (with two minor changes): > > <snip my assumptions> > > Given the problem statement, do you agree that /each/ of these > > assumptions, /on its own/, is reasonable and not just some arbitrary > > statement plucked out of thin air? > > > > If not, which assumption(s) is(are) not reasonable or is(are) > > unneccessarily arbitrary? > > Your assumptions seem consistent with the following formulation of the > problem. > > For n = 1,2,..., define > > A_n = -1/floor((n+9)/10), > R_n = -1/n. > > 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(0)? > Yes; and in fact I chose (1)-(8) to be consistent with just about any sensible interpretation of the problem as given. But I'm currently trying with Tony to completely avoid numerical arguments such as the above, which rely on a complicated definition of "the number of balls at time t", in favor of the much simpler to agree with statement "either there is a ball in the vase at time t, or the number of balls in the vase at time t is 0; and not both". I mean, given his confusion over simple logical arguments like "If (A->not A), then not A", I shudder to think what subtle misunderstandings exist in his version of "define a sequence of functions indexed by n in N". > > > I said that any specific ball was obviously out of the vase at noon. > > > > That's good: we at least agree that it logically follows from (1) - (8) > > that there are no labelled balls in the vase at t=0. > > > > What I honestly find baffling is your repeated claim that it doesn't > > then logically follow from assumptions (2) and (5), that if a ball is > > in the vase at /any/ time, it is a ball which is labelled with a > > natural number; and so therefore the above statement is logically > > equivalent to "there are no balls in the vase at t=0". > > It is rather amazing. It's also sort of fascinating - how can one /not/ understand the argument, and yet give the impression of understanding /some/ sort of logic? It's like some sort of mental blind spot. > The logic seems to be that the limit of the number > of balls in the vase as we approach noon is infinity, so the number of > balls in the vase at noon must be infinity, but all numbered balls have > been removed, therefore the infinity of balls in the vase at noon aren't > numbered. It does have a sort of surreal appeal. If we assume at the start that the number of balls at t=0 is /anything but/ 0 (as TO apparantly does, although he has yet to realize it), then pretty much anything goes. Let your imagination roam! There are a prime number of cubical balls in the vase at noon! ZFC is inconsistent! Cantor is alive and living in Brooklyn New York! I am the current King of France! Cheers - Chas
From: cbrown on 23 Oct 2006 17:29 stephen(a)nomail.com wrote: > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > cbrown(a)cbrownsystems.com wrote: > >> Tony Orlow wrote: <snip> > >> > I said that any specific ball was obviously out of the vase at noon. > >> > >> That's good: we at least agree that it logically follows from (1) - (8) > >> that there are no labelled balls in the vase at t=0. > >> > >> What I honestly find baffling is your repeated claim that it doesn't > >> then logically follow from assumptions (2) and (5), that if a ball is > >> in the vase at /any/ time, it is a ball which is labelled with a > >> natural number; and so therefore the above statement is logically > >> equivalent to "there are no balls in the vase at t=0". > > > It is rather amazing. The logic seems to be that the limit of the number > > of balls in the vase as we approach noon is infinity, so the number of > > balls in the vase at noon must be infinity, but all numbered balls have > > been removed, therefore the infinity of balls in the vase at noon aren't > > numbered. It does have a sort of surreal appeal. > > With the added surreal twist that the limit of the number > of unnumbered balls in the vase as we approach noon is 0, > but the number of unnumbered balls in the vase at noon is > infinite. :) > I think his response, when I pointed this out to him, was either "Oh, shut up!" or "Whatever." Cheers - Chas
From: cbrown on 23 Oct 2006 17:37 stephen(a)nomail.com wrote: > imaginatorium(a)despammed.com wrote: > > > Tony Orlow wrote: > >> David Marcus wrote: > >> > Tony Orlow wrote: > >> > > >> >> At every time before noon there are a growing number of balls in the > >> >> vase. The only way to actually remove all naturally numbered balls from > >> >> the vase is to actually reach noon, in which case you have extended the > >> >> experiment and added infinitely-numbered balls to the vase. All > >> >> naturally numbered balls will be gone at that point, but the vase will > >> >> be far from empty. > >> > > >> > By "infinitely-numbered", do you mean the ball will have something other > >> > than a natural number written on it? E.g., it will have "infinity" > >> > written on it? > >> > > >> > >> Yes, that is precisely what I mean. If the experiment is continued until > >> noon, so that all naturally numbered balls are actually removed (for at > >> no finite time before noon is this the case), then any ball inserted at > >> noon must have a number n such that 1/n=0, which is only the case for > >> infinite n. If the experiment does not go until noon, not all naturaly > >> numbered balls are removed. If it does, infinitely-numbered balls are > >> inserted. > > <snip> > > > Also, suppose for the sake of argument, that there _are_ these > > "infinitely numbered" balls. Are you saying that there is a point at > > which all of the "finitely numbered" balls have been removed (leaving > > the vase empty, which isn't what you are hoping for)? Or are you saying > > there comes a point at which a ball with a number "near the end" of the > > pofnats is being removed, and at the same time the balls being put in > > are actually "infinitely numbered"? That appears to imply that there > > exists a pofnat, call it B, such that B is finite, but 10*B is > > infinite. Is that right? How does this square with even your confused > > understanding of the Peano axioms? > > > Brian Chandler > > http://imaginatorium.org > > Also, supposing for the sake of argument that there are "infinitely > number balls", if a ball is added at time -1/(2^floor(n/10)), and removed > at time -1/(2^n)), then the balls added at time t=0, are those > where -1/(2^floor(n/10)) = 0. But if -1/(2^floor(n/10)) = 0 > then -1/(2^n) = 0 (making some reasonable assumptions about how arithmetic > on these infinite numbers works), so those balls are also removed at noon and > never spend any time in the vase. That coincides with my vision of "T-numbers" as simply the closure of the ordered field of reals with a new number B appended, where B is greater than any real number. It /still/ doesn't imply that 1/B = 0, which is what he needs. Cheers - Chas
From: MoeBlee on 23 Oct 2006 17:45 Tony Orlow wrote: > It has absolutely nothing to do with "cardinality" that I can see. Okay, good, so we're in general accord on the matter. MoeBlee
From: David Marcus on 23 Oct 2006 18:18
Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> Either something happens an noon, or it doesn't. Where do you stand on > >>>> the matter? > >>> What does "something happens" mean, please? I really don't know what you > >>> mean. > >> ??? Do you live in the universe, or in a static picture? When "something > >> happens" o an object, some property or condition of it "changes". That > >> occurs within some time period, which includes at least one moment. > >> There is no moment in this problem where the vase is emptying, > >> therefore, that never "occurs". If you are going to insist that time is > >> a crucial element of this problem, then you should at least be familiar > >> with the fact that it's a continuum, and that events occurs within > >> intervals of that continuum. > > > > Thanks. That explains what "something happens" means. Now, please > > explain what "emptying" means. > > "Empty" means not having balls. To become empty means there is a change > of state in the vase ("something happens" to the vase), from having > balls to not having balls. > > Now, when does this moment, or interval, occur? A reasonable question. Before I answer it, let me ask you a question. Suppose I make the following definitions: For n = 1,2,..., define A_n = -1/floor((n+9)/10), R_n = -1/n. 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). Then V(-1) = 1 and V(0) = 0. If we consider V to be a function of time, at what time does it become zero? -- David Marcus |