An uncountable countable set
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From: Randy Poe on 28 Oct 2006 16:41 Tony Orlow wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Virgil wrote: > >>> In article <4542201a(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> cbrown(a)cbrownsystems.com wrote: > >>>>> When you say "noon doesn't occur"; I think "he doesn't accept (1): by a > >>>>> time t, we mean a real number t" > >>>> That doesn't mean t has to be able to assume ALL real numbers. The times > >>>> in [-1,0) are all real numbers. > >>> By what mechanism does TO propose to stop time? > >> By the mechanism of unfinishablility. > > > > But that's why I asked you a question about variables labelling > > times yesterday, when noon clearly occurred. > > > The experiment occurred in [-1,0). Talk of time outside that range is > irrelevant. Times before that are imaginary, and times after that are > infinite. Only finite times change anything, so if something changes, > it's at a finite, negative time. > > > > > I can define a list of times t_n = noon yesterday - 1/n seconds, > > for all n=1, 2, 3, ... > > Are there balls in the vase for t<-1? No. > > > > > Clearly this list of times has no end. But didn't noon happen? > > Nothing happened at noon to empty the vase, and your desired miracle of > emptiness did not happen before noon, so it had not occurred by noon, so > it was not the case that the vase was empty then, at noon. Comprende? Ay > Que prungisimo! > > > > > How does my list affect the existence of noon yesterday? It's > > unfinishable. Why don't your time-stopping rules work retroactively? > > What's different about this set of times t_n and the set of times > > in the balls-and-vase problem? Why does assigning these labels > > to times yesterday not affect anything, but if I assign those > > labels today, it stops time? > > > > Why, oh why, do the constraints of the problem have to matter? Where do you see "there is no noon" as a constraint in the original problem? It defines a set of events at particular discrete times. That doesn't stop other times from existing. > Why, oh > why, must we alway mean a continuum when we call something continuous, > when I want to declare a discontinuity at convenient places like 0? I have no idea what you're talking about now. > Why > can't we specify what happens at every moment y between x and z, We haven't specified what happens at every moment between x and z. We haven't specified what happens between -1/2 and -1/3 for instance. There are infinitely many other times we haven't specified. If we want to deduce what the state is at -0.4, we'll have to deduce that from the discrete information we have been given. > but > that something totally different is the case at z, without anything > changing it? If we want to deduce what the state is at any time t, we'll have to deduce it from the information given about times <t. That is the case for t=0 as well as for t= -0.4, or for t = -1000. The fact that the problem does not specify an event at t=-0.4, t=-1000, or t=0 is not taken as an assumption that those times never occurred. - Randy
From: Virgil on 28 Oct 2006 16:45 In article <45439300(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> David Marcus wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> David Marcus wrote: > >>>>>>>>> Tony Orlow wrote: > >>>>>>>>>> Mike Kelly wrote: > >>>>>>>>>>> Now correct me if I'm wrong, but I think you agreed that every > >>>>>>>>>>> "specific" ball has been removed before noon. And indeed the > >>>>>>>>>>> problem > >>>>>>>>>>> statement doesn't mention any "non-specific" balls, so it seems > >>>>>>>>>>> that > >>>>>>>>>>> the vase must be empty. However, you believe that in order to > >>>>>>>>>>> "reach > >>>>>>>>>>> noon" one must have iterations where "non specific" balls without > >>>>>>>>>>> natural numbers are inserted into the vase and thus, if the > >>>>>>>>>>> problem > >>>>>>>>>>> makes sense and "noon" is meaningful, the vase is non-empty at > >>>>>>>>>>> noon. Is > >>>>>>>>>>> this a fair summary of your position? > >>>>>>>>>>> > >>>>>>>>>>> If so, I'd like to make clear that I have no idea in the world > >>>>>>>>>>> why you > >>>>>>>>>>> hold such a notion. It seems utterly illogical to me and it > >>>>>>>>>>> baffles me > >>>>>>>>>>> why you hold to it so doggedly. So, I'd like to try and > >>>>>>>>>>> understand why > >>>>>>>>>>> you think that it is the case. If you can explain it cogently, > >>>>>>>>>>> maybe > >>>>>>>>>>> I'll be convinced that you make sense. And maybe if you can't > >>>>>>>>>>> explain, > >>>>>>>>>>> you'll admit that you might be wrong? > >>>>>>>>>>> > >>>>>>>>>>> Let's start simply so there is less room for mutual > >>>>>>>>>>> incomprehension. > >>>>>>>>>>> Let's imagine a new experiment. In this experiment, we have the > >>>>>>>>>>> same > >>>>>>>>>>> infinite vase and the same infinite set of balls with natural > >>>>>>>>>>> numbers > >>>>>>>>>>> on them. Let's call the time one minute to noon -1 and noon 0. > >>>>>>>>>>> Note > >>>>>>>>>>> that time is a real-valued variable that can have any real value. > >>>>>>>>>>> At > >>>>>>>>>>> time -1/n we insert ball n into the vase. > >>>>>>>>>>> > >>>>>>>>>>> My question : what do you think is in the vase at noon? > >>>>>>>>>> A countable infinity of balls. > >>>>>>>>> So, "noon exists" in this case, even though nothing happens at > >>>>>>>>> noon. > >>>>>>>> Not really, but there is a big difference between this and the > >>>>>>>> original > >>>>>>>> experiment. If noon did exist here as the time of any event > >>>>>>>> (insertion), > >>>>>>>> then you would have an UNcountably infinite set of balls. > >>>>>>>> Presumably, > >>>>>>>> given only naturals, such that nothing is inserted at noon, by noon > >>>>>>>> all > >>>>>>>> naturals have been inserted, for the countable infinity. Then > >>>>>>>> insertions > >>>>>>>> stop, and the vase has what it has. The issue with the original > >>>>>>>> problem > >>>>>>>> is that, if it empties, it has to have done it before noon, because > >>>>>>>> nothing happens at noon. You conclude there is a change of state > >>>>>>>> when > >>>>>>>> nothing happens. I conclude there is not. > >>>>>>> So, noon doesn't exist in this case either? > >>>>>> Nothing happens at noon, and as long as there is no claim that > >>>>>> anything > >>>>>> happens at noon, then there is no problem. Before noon there was an > >>>>>> unboundedly large but finite number of balls. At noon, it is the same. > >>>>> So, noon does exist in this case? > >>>> Since the existence of noon does not require any further events, it's a > >>>> moot point. As I think about it, no, noon does not exist in this problem > >>>> either, as the time of any event, since nothing is removed at noon. It > >>>> is also not required for any conclusion, except perhaps that there are > >>>> uncountably many balls, rather than only countably many. But, there are > >>>> only countably many balls, so, no, noon is not part of the problem here. > >>>> As we approach noon, the limit is 0. We don't reach noon. > >>> To recap, we add ball n at time -1/n. We don't remove any balls. With > >>> this setup, you conclude that noon does not exist. Is this correct? > >> I conclude that nothing occurs at noon in the vase, and there are > >> countably, that is, potentially but not actually, infinitely many balls > >> in the vase. No n in N completes N. > > > > Sorry, but I'm not sure what you are saying. Are you saying that what I > > wrote is correct or are you saying it is not correct? I'll repeat the > > question: > > > > We add ball n at time -1/n. We don't remove any balls. With > > this setup, you conclude that noon does not exist. Is this correct? > > Please answer "yes" or "no". > > > > What do YOU mean by "exist"? What does TO mean by "not exist"? When one hypothesizes a time stream, as the gedankenexperiment does, one is presuming a priori that all relevant instants in that time stream will "exist". Not because any particular events are occurring at every instant, but because time is an unbounded continuum, like the set of real numbers. If TO is unhappy, let is modify the original experiment to add a cube to the vase at time noon plus one minute. Now noon has to exist as there are events both before and after it. What is now the state of the vase between noon and the addition of the cube?
From: Virgil on 28 Oct 2006 16:47 In article <4543931c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Please answer "yes" or "no". > > > > By the way, yes and no. TO gets it wrong again. '"Yes" or "no"' is a barely defensible answer, but 'yes and no' is not.
From: Virgil on 28 Oct 2006 16:50 In article <45439360(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: > > Tony Orlow <tony(a)lightlink.com> wrote: > >> stephen(a)nomail.com wrote: > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>> stephen(a)nomail.com wrote: > >>>>> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >>>>>> Tony Orlow wrote: > >>>>>>> David Marcus wrote: > >>>>>>>> Your question "Is there a smallest infinite number?" lacks context. > >>>>>>>> You > >>>>>>>> need to state what "numbers" you are considering. Lots of things can > >>>>>>>> be > >>>>>>>> constructed/defined that people refer to as "numbers". However, > >>>>>>>> these > >>>>>>>> "numbers" differ in many details. If you assume that all subjects > >>>>>>>> that > >>>>>>>> use the word "number" are talking about the same thing, then it is > >>>>>>>> hardly surprising that you would become confused. > >>>>>>> I don't consider transfinite "numbers" to be real numbers at all. I'm > >>>>>>> not interested in that nonsense, to be honest. I see it as a dead > >>>>>>> end. > >>>>>>> > >>>>>>> If there is a definition for "number" in general, and for "infinite", > >>>>>>> then there cannot both be a smallest infinite number and not be. > >>>>>> A moot point, since there is no definition for "'number' in general", > >>>>>> as > >>>>>> I just said. > >>>>>> -- > >>>>>> David Marcus > >>>>> A very simple example is that there exists a smallest positive > >>>>> non-zero integer, but there does not exist a smallest positive > >>>>> non-zero real. If someone were to ask "does there exist a smallest > >>>>> positive non-zero number?", the answer depends on what sort > >>>>> of "numbers" you are talking about. > >>>>> > >>>>> Stephen > >>>> Like, perhaps, the Finlayson Numbers? :) > >>> If they were sensibly defined then sure you could talk about them. > >>> Nothing Ross has ever said has made any sense to me, and > >>> I severely doubt there is any sense to it, but I could be wrong. > >>> The point is, there are different types of numbers, and statements > >>> that are true of one type of number need not be true of other > >>> types of numbers. > >>> > >>> Stephen > > > >> Well, then, you must be of the opinion that set theory is NOT the > >> foundation for all mathematics, but only some particular system of > >> numbers and ideas: a theory. That's good. > > > > You do not understand what people mean when they say set theory > > is the foundation for all mathematics, do you? Set theory > > provides a set of primitives which can be used to describe > > mathematics. Integers, real numbers, hyperreal numbers, imaginary > > numbers, polynomicals, limits, functions, can all be described in > > terms of set theory. Set theory is like assembly language. You can > > use it to build up higher level concepts. Is it the only possible > > foundation for mathematics? Of course not, but it currently appears to > > be the best. > > > > Stephen > > > > "Best" in which respect? Most ridiculous when it comes to oo? Yes, it's > a riot. TO certainly has not come up with anything anywhere nearly as good. All of TO's suggestions to date have been far worse, mostly totally unworkable, and none logically justifiable, or at least none logically justified.
From: David Marcus on 28 Oct 2006 16:55
Virgil wrote: > In article <45435a9c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > David Marcus wrote: > > > Tony Orlow wrote: > > >> MoeBlee wrote: > > >>> Tony Orlow wrote: > > >>>> Eat me. Do you maintain that the two theories are compatible with each > > >>>> other? Is there, and also not, a smallest infinity. > > >>> They're not in conflict, becuase 'smallest infinite' means something > > >>> DIFFERENT in the different contexts. How many times will I say that > > >>> while you STILL refuse to hear it? > > >> So, either smallest has two meanings, or infinite has tow meanings, or > > >> both. Would you like to elucidate the matter by enumerating the various > > >> definitions of "small" and "infinite"? A table might be nice... > > > > > > As many have said, "infinite" has many meanings. I'm afraid it isn't > > > practical to produce a table. > > > > How about a list? ;) > > As lista are merely function having N as domain, and functions can often > be given by formulae, it is easy to produce some lists. > > For example, f:N -> R: n |-> n, is a simple list. Tony wants a list of all the meanings of the word "infinite" in mathematics. -- David Marcus |