An uncountable countable set
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From: Tony Orlow on 27 Oct 2006 19:45 Lester Zick wrote: > 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~~ The positive reals>0 include elements less than any positive natural>0, an infinite number in (0,1).
From: David Marcus on 27 Oct 2006 19:54 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> stephen(a)nomail.com wrote: > >>>>> What are you talking about? I defined two sets. There are no > >>>>> balls or vases. There are simply the two sets > >>>>> > >>>>> IN = { n | -1/(2^floor(n/10)) < 0 } > >>>>> OUT = { n | -1/(2^n) < 0 } > >>>> For each n e N, IN(n)=10*OUT(n). > >>> Stephen defined sets IN and OUT. He didn't define sets "IN(n)" and "OUT > >>> (n)". So, you seem to be answering a question he didn't ask. Given > >>> Stephen's definitions of IN and OUT, is IN = OUT? > >> Yes, all elements are the same n, which are finite n. There is a simple > >> bijection. But, as in all infinite bijections, the formulaic > >> relationship between the sets is lost. > > > > Just to be clear, you are saying that |IN - OUT| = 0. Is that correct? > > (The vertical lines denote "cardinality".) > > Um, before I answer that question, I think you need to define what you > mean by "|IN - OUT|" =0. How are you measuring IN and OUT, and how do > you define '-' on these "numbers"? IN and OUT are sets, not "numbers". For any two sets A and B, the difference, denoted by A - B, is defined to be the set of elements in A that are not in B. Formally, A - B := {x| x in A and x not in B} Note that the difference of two sets is again a set. For any set, the notation |A| means the cardinality of A. So, saying that |A| = 0 is equivalent to saying that A is the empty set. In particular, for any set A, we have |A - A| = 0. -- David Marcus
From: stephen on 27 Oct 2006 19:54 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
From: Tony Orlow on 27 Oct 2006 19:55 cbrown(a)cbrownsystems.com wrote: > 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". > What changes the "number" of balls, if not "additions" and "subtractions"? >>>> 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. And, what if, for all t in [t1,t0), there are balls? Then balls can only disappear entirely at t0. > > Compare (T1) and (*); they say different things. And? Not so different after all, eh? > >>> 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. Then a dozen more explanations are a waste of time. "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/?" > Sure, the question is put in terms proscribed by the constraints of the problem. Nothing can occur at noon. It cannot become empty then, nor before. >> 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. Then you admit you have no answer, given your formulation? > > 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. So, you don't want to assume that the only way balls can leave the vase is by removal, or that the vase can only become empty by balls leaving it? Which of those would you like to reject? > > 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
From: Virgil on 27 Oct 2006 19:55
In article <454298b7(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Ahem. I said that Robinson's analysis seems to have nothing to do with > >> transfinitology. > > > > You agreed it is not an ordering by cardinality. > > Yes, and you asked me what I would think if it was derived from > transfinite set theory. You're tripping over your tail. When TO can explain how Robinson's ultrafilters can work without at least some uncountably infinite sets, TO will perhaps stop tripping over his own tail. > And the use of the > > word 'infinite' is A DIFFERENT SENSE in the two different contexts. So > > there is NO CONTRADICTION. > > OHHHHH!!!!!!! WOW!!!!! NO CONTRADICTION!!!!! WITHIN MATHEMATICS!!! > THAT'S GREAT!!!!! WAS THAT THE PHONE????? No! That ringing in TO's ears is merely the reverberations through his empty head caused by his shouting. > > Read Robinson for real and then comment. See how quickly you can absorb > his actual section on logical tools, and whether you might skip ahead > and then back. I have read it enough to know that TO has not understood whatever part of it he claims to have read. > > You don't learn the > > mathematical logic and set theory that are the basis for the material > > and even pretty much ignore the mathemtatical logic and set theory that > > the author himself summarizes in the book. (You need to start by > > learning how to work in the predicate calculus), and (2) you don't > > listen when someone tries to warn you about the confusions you are > > making due to your not understanding the basis and context. > > > > Well, it doesn't help when they freak out because they can't handle > questioning certain iffy logical constructs. You're getting a tad edgy > lately, which I understand. I'm not floating on a cloud myself. TO is in his own cloud cuckooland, but whether he is floating or sinking is not apparent from within reality. > > Why does the very mention of no smallest infinite elicit such bile, > then? If TO wants to avoid that "bile" he can merely stop claiming that there is no smallest infinite ordinal. > > 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... A head might be nice, too. Too bad TO doesn't have one. |