An uncountable countable set
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From: David R Tribble on 27 Oct 2006 14:02 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. > David R Tribble wrote: >> Of course there is a "something" between "before noon" and "noon" where >> each ball disappears. At step n, time 2^-n min before noon, ball n is >> removed. This happens for every ball, since there is a step n for >> every ball. The balls are removed, one by one, one at each step, >> before noon. > Tony Orlow wrote: > As each ball n is removed, how many remain? Can any be removed and leave > an empty vase? Each ball n is placed into the vase at time 2^int(n/10), and then later removed at time n. This happens for every ball before noon. So every ball is inserted and then later removed from the vase before noon. At any given time n before noon, ten balls are added to the vase and then ball n (which was added to the vase in a previous step) is removed. Your entire confusion results from assuming a "last" time prior to noon, but there is no such time.
From: Lester Zick on 27 Oct 2006 14:05 On Fri, 27 Oct 2006 16:30:04 +0000 (UTC), 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. So non zero integers are not real? Or is this another zenmath conundrum? Just curious. > 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 ~v~~
From: cbrown on 27 Oct 2006 14:07 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > >> Mike Kelly wrote: > > > > <snip> > > > >>> My question : what do you think is in the vase at noon? > >>> > >> A countable infinity of balls. > >> > >> This is very simple. Everything that occurs is either an addition of ten > >> balls or a removal of 1, and occurs a finite amount of time before noon. > >> At the time of each event, balls remain. At noon, no balls are inserted > >> or removed. > > > > No one disagrees with the above statements. > > > >> The vase can only become empty through the removal of balls, > > > > Note that this is not identical to saying "the vase can only become > > empty /at time t/, if there are balls removed /at time t/"; which is > > what it seems you actually mean. > > > > This doesn't follow from (1)..(8), which lack any explicit mention of > > what "becomes empty" means. However, we can easily make it an > > assumption: > > > > (T1) If, for some time t1 < t0, it is the case that the number of balls > > in the vase at any time t with t1 <= t < t0 is different than the > > number of balls at time t0, then balls are removed at time t0, or balls > > are added at time t0. > > > > Well, you have (8), which is kind of circular, but related. (8) simply states that if there are no balls in the vase at time t, then the vase is empty at time t; and if the there is a ball in the vase at time t, then the vase is not empty at time t. It states nothing about "how that event occurs". > > >> so if no balls are removed, the vase cannot become empty at noon. It was > >> not empty before noon, therefore it is not empty at noon. Nothing can > >> happen at noon, since that would involve a ball n such that 1/n=0. > > > > Now your logical argument is complete, assuming we also accept > > (1)..(8): If the number of balls at time t = 0, then by (7), (5) and > > (6), the number of balls changes at time 0; and therefore by (T1), > > balls are either placed or removed at time 0, implying by (5) and (6) > > that there is a natural number n such that -1/n = 0; which is absurd. > > Therefore, by reductio ad absurdum, the number of balls at time 0 > > cannot be 0. > > > > However, it does not follow that the number of balls in the vase is > > therefore any other natural number n, or even infinite, at time 0; > > because that would /equally/ require that the number of balls changes > > at time 0, and that in turn requires by (T1) that balls are either > > added or removed at time 0; and again by (5) or (6) this implies that > > there is a natural number n with -1/n = 0; which is absurd. So again, > > we get that any statement of the form "the number of balls at time 0 is > > (anything") must be false by reductio absurdum. > > > > So if we include (T1) as an assumption as well as (1)..(8), it follows > > logically that the number of balls in the vase at time 0 is not > > well-defined. > > That is correct. Noon is incompatible with the problem statement. > > > > > Of course, we also find that by (1)..(8) and (T1), it /still/ follows > > logically that the number of balls in the vase at time t is 0; and this > > is a problem: we can prove two different and incompatible statements > > from the same set of assumptions > > Right. Your conclusion is at odds with the notion that only removals may > empty the vase, which seems to be an obvious assumption, no other means > of achieving emptiness having been mentioned. > But (T1) does /not/ merely state "Only removals may empty the vase". (T1) states something quite a bit stronger: it states that if the vase becomes empty /at time t/ then removals occur /at time t/. I would formalize "only removals may empty the vase" (which I agree is a desirable assumption) as: (*) If, for some time t1 < t0, it is the case that the number of balls in the vase at time t1 is different than the number of balls at time t0, then there is some time t with t1 <= t <= t0, such that balls are removed at time t, or balls are added at time t. Compare (T1) and (*); they say different things. > > > > So at least one of the assumptions (1)..(8) and (T1) must be discarded > > if we are to resolve this. What do you suggest? Which of (1)..(8) do > > you want discard to maintain (T1)? > > I don't believe any of those assumptions are the problem. (2) should > state that t<t0, not t<=t0, at any event. But, that's irrelevant. The > unspoken assumption on your part which causes the problem is that noon > is part of the problem. I don't understand this complaint. "Noon exists" follows from (1): when we we speak of the time "noon", we mean the real number 0. Do you claim that the real number 0 does not exist? And certainly noon is "part of the problem": the original problem explicitly asks: "What is the number of balls in the vase /at noon/?" > Clearly, it cannot be, because anything that > happened at t=0 would involve n s.t. 1/n=t. Essentially, the problem > produces a paradox by asking a question which contradicts the situation. This doesn't imply that noon "doesn't occur" - it simply states that "the number of balls in the vase at noon" cannot be determined in a well-defined manner consistent with our assumptions. But that is only the case if we assume (T1). If we /don't/ assume (T1), or we instead assume (*), then your statement does not follow; instead it follows that the vase is empty at noon. And if we /do/ accept (T1), we still have the problem I alluded to: we can /still/ prove from (1)..(8) that the problem is well-defined (empty vase at noon); but we can also prove that the problem is /not/ well-defined by the argument you give above. So something is not right with at least one of our assumptions; and the usual approach is to abandon (T1) in favor of (*). The situation here is similar to the problem: "Is Socrates mortal?" If we agree with /all/ of the following assumptions: (a) Socrates is a man. (b) All men are mortal. (c) No mortal lives forever. (d) Socrates lives forever through his writings. Then we can prove by (a) and (b) that Socrates is mortal; but we can also prove that Socrates is not mortal by (c) and (d). /None/ of (a)..(d) are stated /explicitly/ in the problem. A valid argument either way must be based on a /non-contradictory/ set of assumptions. So at least one of (a)..(d) must be discarded before we can claim
From: cbrown on 27 Oct 2006 14:18 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: <snip> > > A poet would say that "A rose is still a rose by any other name"; a > > mathematician would say that "By 'a rose' we mean a repesentative of an > > equivalence class of those herbacious plants having the following > > properties: thorns, leaves found on alternating sides of the stem; > > flowers having a a sweet smell, vaselike growth pattern, ... From this, > > we can deduce that the assertion of the heavy metal ballad, 'Every rose > > has its thorn', logically follows." > > > > Sometimes these different modes of thinking overlap; but more often, > > they lead to different conclusions about what is or isn't the state of > > affairs. > > Very true, but like the Zen archer, we have to train our intuitions, and > when they are in harmony with the universe, the arrow hits its mark.:) > In physics those intuitions are the ones which accord with "the universe" of real world measurements. In mathematics, those intuitions are the ones which are instead in accord with "the universe" of logical conclusions from agreed upon premises. The archers are aiming at different targets; so they develop different intuitions. Cheers - Chas
From: David Marcus on 27 Oct 2006 14:23
Lester Zick wrote: > On Fri, 27 Oct 2006 16:30:04 +0000 (UTC), stephen(a)nomail.com wrote: > >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. > > So non zero integers are not real? That's a pretty impressive leap of illogic. > Or is this another zenmath conundrum? Just curious. -- David Marcus |