An uncountable countable set
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From: Virgil on 23 Oct 2006 17:02 In article <453d16f3(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> In Chapter III, section 3.1.1, he states: > >> > >> "There is no smallest infinite number. For if a is infinite then a<>0, > >> hence a=b+1 (the corresponding fact being true in N). But b cannot be > >> finite, for then a would be finite. Hence, there exists an infinite > >> numbers [sic] which is smaller than a." > > > > I'll try to take a look at the book soon, but I very strongly suspect > > that he's using 'infinite' not to refer to cardinality, but rather to > > position in certain orderings. That is fine, as long as we understand > > the terminology in the context. As far as I know, it doesn't contradict > > that there is a least infinite cardinality. > > > > MoeBlee > > > > It has absolutely nothing to do with "cardinality" that I can see. He > defines an infinite element of *N as being "larger than any finite > number", such that x e N ^ y e *N and ~y e N -> y>x. N is a subset of *N. But none of Robinson's non-standard numbers are cardinalities.
From: cbrown on 23 Oct 2006 17:05 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > >> cbrown(a)cbrownsystems.com wrote: > >>> Tony Orlow wrote: > >>>> cbrown(a)cbrownsystems.com wrote: > >>>>> Tony Orlow wrote: > >>>>>> cbrown(a)cbrownsystems.com wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> cbrown(a)cbrownsystems.com wrote: > >>>>>>>>> Tony Orlow wrote: > > <snip> > >>> I attempted to describe those assumptions in my previoius post. Did you > >>> read those assumptions? If so, do you agree with those assumptions? > >> At this point I don't recall your previous post. I've been off a bit. > > > > Well, allow me to repeat them here (with two minor changes): > > > > In order to interpret the problem > > > > "At each time t = -1/n where n is a (strictly positive) natural number, > > we place the balls labelled 10*(n-1)+1 through 10*n inclusive in the > > vase, and remove the ball labelled n from the vase. What is the number > > of balls in the vase at time t=0?" > > > > I make the following simple (and I would claim, fairly uncontroversial > > and natural) assumptions: > > > > --- (object permanence) > > > > (1) When we speak of a time t, we mean some real number t. > > > > (2) If a ball is in the vase at any time t0, there is a time t <= t0 > > for which we can say "that ball was placed in the vase at time t". > > > > (3) If a ball is placed in the vase at time t1 and it is not removed > > from the vase at some time t where t1 <= t <= t2, then that ball is in > > the vase at time t2. > > > > (4) If a ball is removed from the vase at time t1, and there is no time > > t such that t1 < t <= t2 when that ball is placed in the vase, then > > that ball is not in the vase at time t2. > > > > ---- (obedience to the problem constraints) > > > > (5) If a ball is placed in the vase at some time t, it must be in > > accordance with the description given in the problem: it must be a ball > > with a natural number n on it, and the time t at which it is placed in > > the vase must be -1/floor(n/10). > > > > (6) If a ball is removed from the vase at some time t, it must be in > > accordance with the description given in the problem: it must be a ball > > with a natural number n on it, and the time t at which it is removed > > from the vase must be -1/n. > > > > (7) If n is a natural number with n > 0, then the ball labelled n is > > placed in the vase at some time t1; and it is removed from the vase at > > some time t2. > > > > --- (very general definition of "the vase is empty at noon") > > > > (8) the number of balls in the vase at time t=0 is 0 if, and only if, > > the statement "there is a ball in the vase at time t=0" is false. > > > > --- > > > > Perhaps you would add other assumptions (9), (10), etc.; but my > > question is: > > > > 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? > > > > <snip> > > Those all look reasonable to me as I read them. Good. I propose that if we agree that these statements are true, then we also agree that any logical implications that follow from, these statements we also agree to be true. > I don't see any > statement regarding the fact that ten balls are added for every one > removed, though that can be surmised from the insertion and removal > schedule. It certainly can be deduced from (1)..(8). But it is worth noting that it was already the case that the statement "for /every/ 10 balls added, 1 ball is removed" is not /explicitly/ stated at any point the original problem statement. It is /already/ something we had to deduce - for example, by the (trivial, but neccessary) logical argument that if t is a time when any balls are added, then t is a time that 10 balls are added and also a time when a ball is removed, But we should be careful regarding what we are really claiming in this case. Equally, the statement "for every ball that is added, 1 ball is removed" can also be deduced from (7), (5) and (6) (in this case we don't require that the adding and removing occur at the same time t). > That's the salient fact here. Regardless of its saliency, we can at least agree that it follows logically from (1)-(8). > You never remove as many as you > add, so you can't end up empty. That is not a conclusion we can draw from (1)..(8). Instead, this is the unstated assumption to which you return again and again: (Proposition T) If you never remove as many as you add, then there is a ball in the vase at time 0. If we simply /require/ (T) to be true, and damn the consequences, then of course it /is/ true, That is what we mean when we say "that argument is circular". But if we say "maybe it's true, maybe it's not true", then what we would like to find is a proof of (T) that doesn't rely on simply /assuming/ that it is true; or a proof that (T) is false, that doesn't rely on simply /assuming/ that it is false. In other words, we need a series of statements that follow from some set of assumptions like (1)..(8) that doesn't include "(T) is true", that result in the conclusion "therefore (T)". That is exactly what is required to say "We have proven (T) from (1)..(8)". Alternatively, we need a series of statements that follow from some set of assumptions like (1)..(8) that doesnt include "(T) is false" that result in the conclusion "therefore, (T) is false". That is all that is required to say "We have proven that (T) is false from (1)..(8)". The latter is what I provided in my argument at the end of my previous post. > > > > >>>> You claim nothing changes at noon. > >>> Where, exactly, above do I claim that "nothing changes at noon"? > >> Do you disagree with the other standard-bearers, and claim that > >> something DOES occur at noon? > > > > That is not a response to /my/ question "where, exactly, above do I > > claim that 'nothing changes at noon'?" > > > > /I/ don't claim that "something occurs at noon"; nor do /I/ claim that > > "nothing occurs at noon". > > Uh, what would be your opinion on the matter. CAN something occur at > noon in this experiment or not? Either way, you have a problem. > In order to answer your question I need to know what you mean, in mathematical terms, by "someth
From: Virgil on 23 Oct 2006 17:13 In article <453d18da(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > >>> What about #5? It says that every ball in the vase has a natural number > >>> on it. Do you agree with that? > >> That is in the problem statement. Therefore, nothing transpires at noon, > >> since -1/n<0 for all n e N. > > > > That statement offers no problems to those who do not require anything > > beyond the conditions of the original problem. > > The problem statement precludes the arrival of noon. Only in TOmania. > > >>>> 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. > > > > In any physical world, something happening, or changing, requires an > > interval of time of strictly positive duration to occur. Nothing can > > "happen" instantaneously in that world. > > But even where infinitely fast processes occur, there has to be at least > a moment where it occurs. > > > > > So what does TO mean by "something happening" instantaneously in a > > mathematical world, for which we have no physical world analog? > > I mean it can happen in a moment, a period of time less than any finite > length. > > > > >> There is no moment in this problem where the vase is emptying, > >> therefore, that never "occurs". > > > > The process of "emptying" may not occur, in the sense of the number of > > balls decreasing from one moment to another at any time before noon, but > > the result does, in the sense of there being no ball which has not been > > removed, at noon. > > > > > > You are not allowed to proceed to noon, or you would have infinite balls > with numbers such that 1/n=0. Sorry, Charlie. In my world, TO's delusions cannot forestall noon arriving on schedule. Regardless of what goes on, or doesn't, in TO's world. > > > > > > >> 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. > > > > Then the whole problem is cooked, since no ball can be inserted or > > removed instantaneously in a physical world as the problem requires, nor > > can we have infinitely many physical balls nor a vase large enough to > > hold them all. > > You cannot proceed to noon without inserting infinite balls, and you > cannot empty the vase before noon. Sorry. TO is wrong about what transpires in my world. He can only speak for what can happen in his. In my world, noon comes with an empty vase. > > > > > If one accepts the conditions of the gedankenexperiment at all, then the > > only conclusion that fits those conditions is that the vase is empty at > > noon. > > No, it is obviously filled with uncountably many infinitely numbered > balls at noon, if one insists against the rules of the gedanken that > noon can occur. In my world, the problem limits us to balls which are given to exist by the problem and times that go smoothly past noon regardless of TO's trepidations. > > > > One is quite free to object to the conditions themselves as being > > unrealistic, but then one cannot also simultaneously accept those > > conditions. > > One can point out the obvious contradiction. While it may be paradoxical, it is not contradictory of anything except the validity of TO's intuitions.
From: Virgil on 23 Oct 2006 17:19 In article <453d193d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > And where do these allegedly infinitely numbered balls come from? > > I do not recall any of them being mentioned in the original gedanken, so > > that TO is creating his own separate gedanken. > > > > Note that whatever TO may require in his version of the gedanken, his > > requirements do not alter that no such thing occurs in the original. > > > > TO takes the childish position that if he cannot have things his own > > way playing by the rules, he will change the rules to get his own way. > > They come from the inclusion of noon in the experiment. If any ball is > removed at noon, ten are inserted, and their numbers will be of a form > that makes them infinite. According to the rules, all insertions and removals occur strictly before noon. So nothing is either inserted or removed AT noon. And none of TO's mythical monstrosities with infinite numberings are either required or allowed. > Nothing occurs at noon in your experiment, and > the vase is empty before noon. Not empty at any time before noon, but empty at noon.
From: Virgil on 23 Oct 2006 17:26
In article <453d1a98(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Randy Poe 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, > >>> The clock ticks till noon and beyond. However, the explicitly- > >>> stated insertion times are all before noon. > >>> > >>>> so that all naturally numbered balls are actually removed (for at > >>>> no finite time before noon is this the case), > >>> There is no finite time before noon when all balls have > >>> been removed. > >>> > >>> However, any particular ball is removed at a finite > >>> time before noon. > >>> > >>>> then any ball inserted at > >>>> noon must have a number n such that 1/n=0, > >>> However, there is no ball inserted at noon. > >>> > >>>> which is only the case for > >>>> infinite n. If the experiment does not go until noon, not all naturaly > >>>> numbered balls are removed. > >>> The experiment goes past noon. No ball is inserted at noon, > >>> or past noon. > >>> > >>> - Randy > >>> > >> Randy, does it not bother you that no ball is removed at noon, > > > > ... I agree with that... > > > >> and yet, when every ball is removed before noon, > > I should have said "each"... > > > > > ... I agree with that... > > > >> balls remain in the vase? > > > > .. I don't agree with that. > > > > When did I ever say balls remain in the vase? Every ball > > is removed before noon. No balls remain in the vase at > > noon. > > Do you disagree with the statement that, at every time -1/n, when ball n > is removed, for every n e N, there remain balls n+1 through 10n, or 9n > balls, in the vase? > > > > >> How do you explain that? > > > > I would certainly have difficulty understanding how the vase > > could be non-empty at noon, given that every ball in the vase > > is removed before noon. > > There is no ball, in all of N, for which the vase is empty at its > departure from the vase. > > > > > But YOU are the one who says the vase is non-empty at > > noon. I never said such a thing. I'm certainly not going to > > defend YOUR illogical position. > > I was saying it is non-empty at every one of the finite times before > noon where any ball is inserted or removed. Do you argue against THAT > statement? > > > > > So you now agree that it makes no sense that the vase > > could be non-empty at noon? That the vase must, in other > > words, be empty? > > > > - Randy > > > > No, you misread. If the vase in not empty at noon, it must contain at least one naturally numbered ball, as these are the only things put into it before noon. If the set of naturally numbered balls in the vase at noon is not empty, there is a smallest numbered ball in that set, since the naturals are well ordered. So if TO claims that the vase is not empty at noon, he must claim there is a natural numbered ball of smallest number in that vase, but cannot name it. We, on the other hand, can easily prove inductively, that the set of numbered balls removed from the vase includes every naturally numbered ball. |