An uncountable countable set
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From: David Marcus on 31 Oct 2006 20:12 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> Randy Poe wrote: > >>> The ORIGINAL problem. This is a new one, inspired by > >>> the original, but it is one with no balls, no vases, no > >>> time steps, no iterations. Just a definition of two subsets > >>> of the natural numbers, one called IN and one called > >>> OUT. > >>> > >>> The definition of the set IN does not include a definition > >>> of something called IN(n). > >>> > >>> You are being asked to characterize these two subsets. > >>> > >>> - Randy > >>> > >> Set-theoretically, they are the same set, by the axiom of > >> extensionality. That doesn't mean the axioms of ZFC account for all the > >> information in the problem. There is the combining of +10 and -1 in each > >> iteration n at time t=-1/n, a coupling that is not being addressed by > >> your method. You are considering the two sets statically, outside of > >> time, as completed, but for any n, the max of in is 10 times the max of > >> out. Just because these both have some limit at oo, even though they > >> don't reach it, doesn't mean they reach it at the same time. They DON'T > >> reach it, and if they did, if noon occurred, in would reach it in 1/10 > >> the time as out. But,there is no such ending to the finites, and so the > >> set-theoretic approach using the set N is bogus at its core. > > > > It is remarkable that you seem to be unable to answer any post without > > mentioning the ball and vase problem. Why is this? Are you afraid that > > if you do, you will be trapped into an inconsistentcy? > > > > Is the following an accurate description of what you are saying? > > > > 1. You agree that (given the definitions above) IN = OUT and that |IN > > \OUT| = 0. > > > > 2. You don't agree that, in the ball and vase problem, the number of > > balls in the vase at noon is |IN\OUT|. > > 2 Are you saying that you only agree with #2? I wasn't suggesting you pick one of the two things I wrote. I thought you would agree to both. They don't contradict each other, so you can logically agree to both. -- David Marcus
From: David Marcus on 31 Oct 2006 20:16 stephen(a)nomail.com wrote: > David R Tribble <david(a)tribble.com> wrote: > > [Apologies if this duplicates previous responses] > > Tony Orlow wrote: > >> However, when talking > >> about processes of adding and removing elements, the sets are not > >> static, but changing with each event. > > > Incorrect. If we define set A as containing the elements a, b, and c, > > then A = {a, b, c}. Period. If we then talk about adding elements d > > and e to set A, we're not actually changing set A, but describing > > another set, call it A2, that is the union of A and {d, e}, so > > A2 = {a, b, c, d, e}. > > > Nothing is ever "added to" a set. Rather, we apply operations (union, > > intersection, etc.) to existing sets to create new sets. We don't > > change existing sets. > > Just like when we add 5 to 2 to get 7, we do not change the 5 or 2 > to create a 7. Or when you celebrate a birthday, your age changes, > but the number that represented your age does not change. A different > number is used to represent your age, but the "old" number remains > as it always ways. That's a good example. > This idea of "changing" sets seems to be at the heart of a lot > of people's misconceptions about set theory. Indeed. People seem to let their intuitive ideas of the words confuse them. In mathematics, a "set" is simply what mathematicians have agreed it to be. This doesn't mean that mathematics can't deal with things that change. It just means you need to use the agreed upon terminology (e.g., "function") if you want to be able to communicate your ideas. -- David Marcus
From: David Marcus on 31 Oct 2006 20:22 imaginatorium(a)despammed.com wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > > > Mike Kelly wrote: > > > > Tony Orlow wrote: > > <snip> > > > > > 2) How come noon "exists" in this experiment but it didn't exist in the > > > > original experiment? Or did you give up on claiming noon doesn't > > > > "exist"? What does that mean, anyway? > > > > > > Nothing is allowed to happen at noon in either experiment. > > > > Nothing "happens" at noon? I take this to mean that there is no > > insertion or removal of balls at noon, yes? Well, I agree with that. > > Hmm. Yes, there is no ball whose insertion time or removal time is > noon. But it seems to me that this "happen" is underdefined in a way > that can cause confusion. Does something "happen" to either of these > functions at x=0: > > f(x) = 1 if x<0 ; 0 if x>=0 > > g(x) = 1 if x<=0 ; 0 if x>0 > > It seems to me that it is true (within the accuracy of normal > communication) to say that both f() and g() "drop from 1 to 0 at x=0" > even though the functions are different. > > Similarly, it seems to me that clearly something "happens" (in any > normal sense) at noon in the standard vase problem - what happens is > that the frenzy of unending sequences of insertion and removal come to > a halt. Good point. -- David Marcus
From: David Marcus on 31 Oct 2006 20:26 Randy Poe wrote: > I think that when Tony and Han say "noon doesn't exist" they > really mean "there is no noon on the clock in that experiment", > as a way of saying "I have no idea how to answer questions about > noon in that experiment, so I'll say that there is no noon and that > way I don't have to answer any such questions." Isn't it simpler? Noon doesn't exist because the number of balls function is discontinuous at noon. Since the number of balls should be a continuous function of time (like the mental picture of water being poured into the vase), if it isn't continuous, then if noon existed, we'd be forced to admit the discontinuity. The only way to avoid it is to deny that the function is defined. Since the vase just vanishing doesn't match the mental picture, we can't say the vase has vanished. But, we can imagine our mental movie ending, in which case noon doesn't exist. > I've asked questions similar to yours. The answer is: "It's easy > for me to figure out there's a ball in the vase at noon. Therefore > I will allow noon to 'exist' in this problem." -- David Marcus
From: David Marcus on 31 Oct 2006 20:29
Mike Kelly wrote: > Tony Orlow wrote: > > Where are the iterations mentioned there? You're missing the crucial > > part of the experiment. By your logic, you could put them in in any > > order and remove them in any order, and when you say both processes are > > done, nothing's left, but that's BS. It ignores the sequence specified. > > This is just a distraction. > > Yes, if you insert and remove exactly the same balls then you get the > same result when you're done, no matter what order you did it all in. > Why is that BS? It seems blindingly obvious. > > But I forgot, you think that if you shift all the insertions 1 minute > further back in time, you DO get an empty vase at noon, right? I really > don't understand how your mind works. Try the mental picture with the water. We fill it up, then we start letting it run out. No reason all the water shouldn't empty out of the vase by noon. -- David Marcus |