An uncountable countable set
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From: Virgil on 17 Sep 2006 16:22 In article <450d5f76(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike, you haven't responded to my use of IFR An IFR, being dependent on order relations, at best measures order relations, not their underlying sets.
From: Virgil on 17 Sep 2006 16:46 In article <450d664f(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Can cardinality be said to be "accurate", in the sense that it detects > all changes in set size? If one only looks at the set itself, yes. The only inherent relation on a set is the identity relation which determines for each pair of members x and y whether x = y or not. Taking only that relation, and no other, into account, cardinality is as accurate as one can get. > >> Cardinality gives a certain gross measure of complexity, but to > >> consider the alephs to be any exact numbers of any sort is > >> unjustified, in my opinion. > > > > You can consider them inexact numbers if you want. This has no effect on > > the mathematical content of the set theoretical theory of cardinality. > > But it does have an effect on how cardinality is treated. Only as it is treated by TO. > > The question is this. If cardinality DOES apply generally, then why does > it contradict so many specific cases? Name one.
From: Virgil on 17 Sep 2006 19:55 In article <450d6bf1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> As far as probability goes, it certainly depends on the concept of > >> sets, since probability more or less measures a subset of events > >> with respect to the entire set of possible events. However, the > >> same question remains as with the rest of transfinitology - is the > >> cardinality generalization, based solely on raw bijection, really > >> the most appropriate generalization from the finite to the > >> infinite for sets? > > > > Irrelevant. > > To what? To the particular question at issue, namely whether a uniform probability distribution is possible on a countably infinite set. > > > > >> Do we need to > >> know the last element and exact range to derive a probability for > >> something as simple as "n is a multiple of 3"? > > > > No, but we need to know that it is possible to define a uniform > > distribution on the set. > > Which requires an average, which requires a range. Neither of which exist for the set of naturals. > > > > >>> It seem your argument is based on the idea that infinites do not exist > >>> in physical reality. But mathematics is abstract, so this seems an > >>> absurd objection. > >> I think if Wolfgang and Han were offered a more sensible treatment of > >> the infinite case, they might find it more palatable. > > > > Uh, like yours you mean? Snicker. Han rejects completely the existence > > of a "completed infinity". This is pretty close to the opposite of what > > you're trying to do. > > That is true, for the reason I just mentioned. The standard streatment > is nonsensical. > > > > >>> If you refuse the idea of infinite sets, what does it mean to you to > >>> say a function has domain and range R? > >>> > >> What does it mean to you, if not that one can use that range as a > >> constant for infinite sets? Why can't we say that, over the entire range > >> of R, the naturals have twice the density of the evens, and so are twice > >> as large a set? > > > > You're free to do so. The evens do have have a density of 1/2 in the > > naturals (not in R, I'm not sure you meant to say R. I think they both > > have 0 density in R...). If you want to think of this as meaning the > > set is twice the "size" then you're free to do so. > > I said "the range of R", meaning over the real line. Since over that alleged "range of R" both densities are zero, their ratio is indeterminate. > > > > >> Why should set theory contradict so basic an understanding? > > > > It doesn't. You really don't get it, do you? I've told you half a dozen > > times : cardinality doesn't claim to be the only or the "correct" > > generalisation of size to infinite sets. > > > > If it claims that the vase is empty, when sequences tell us the vase is > not, then a set is not a generality of a sequence. It is not cardinality which tells us that. When every ball that is put into the vase is also removed from the vase before the vase's contents are analyzed, TO wishes to claim that the vase, when analyzed still holds some of those balls. But declines to say which ones.
From: Tony Orlow on 17 Sep 2006 19:59 Mike Kelly wrote: > Tony Orlow wrote: >> Mike Kelly wrote: >>> Tony Orlow wrote: >>>> Mike Kelly wrote: >>>>> Han.deBruijn(a)DTO.TUDelft.NL wrote: >>>>>> Mike Kelly wrote: >>>>>> >>>>>>> [ ... snip ... ] It's not clear to me that providing finite examples then >>>>>>> saying "obviously this holds for infinite cases too" without any >>>>>>> justification whatsoever should be at all convincing to anyone. >>>>>> It may be not clear to any mathematician, but it is clear to any >>>>>> scientist. The reason is that infinities do not really exist. >>>>>> They only exist as an attempt to make the "very large" rigorous >>>>>> in some sense. The moment you forget this, you get into trouble. >>>> Han, if I prove inductively, say, that 2^x>2*x for all x>2, do you find >>>> it objectionable to say that this also applies to any infinite value, if >>>> such a thing existed, given that any infinite value would be greater >>>> than any finite value, and therefore greater than 2? >>>> >>>>> But we are discussing whether there exists a uniform distribution over >>>>> the naturals. If you don't think this claim means anything at all then >>>>> why do you dispute it? If you reject the existence of the set of >>>>> natural numbers then you reject the set theory probability is based on. >>>>> So why bother to argue against individual theorems? You don't accept >>>>> *any* of probability theory. >>>> Just because someone disagrees with the transfinite portions of set >>>> theory doesn't mean they reject all of set theory. Clearly those of us >>>> who object do so on the basis of the conclusions drawn in infinite case, >>>> which derive from the axiom of infinity and/or the axiom of choice. >>> So, which do you reject? The axiom of infinity or the axiom of choice? >>> >>>> As >>>> far as probability goes, it certainly depends on the concept of sets, >>>> since probability more or less measures a subset of events with respect >>>> to the entire set of possible events. However, the same question remains >>>> as with the rest of transfinitology - is the cardinality generalization, >>>> based solely on raw bijection, really the most appropriate >>>> generalization from the finite to the infinite for sets? >>> Irrelevant. >> To what? Ahem! No answer? To a request for clarification of a curt dismissal? Hmmm... >> >>>> Do we need to >>>> know the last element and exact range to derive a probability for >>>> something as simple as "n is a multiple of 3"? >>> No, but we need to know that it is possible to define a uniform >>> distribution on the set. >> Which requires an average, which requires a range. > > What does "requires an average" mean? It means it requires a count and a sum, and the notion of division. > > Loosely speaking, to define a uniform distribution to select an element > from a set one assigns a contsant probability to each element such that > they all sum to 1. No such contstant exists for countable sets. They cannot exist for "countably infinite" sets, since those have no upper bound (omega notwithstanding). Without an upper bound, there's no mean, and no distribution. Do they exist for "uncountably" (aka actually) infinite sets? Is there an average value of the reals in [0,1]? No, that also would require the conception of a value less than any finite, an infinitesimal probability for each real, which would sum to 1. So, you probably reject that notion as well. However, the average value of the reals in [0,1] is quite obviously 1/2. So, you have a bit of a problem there. Yes, there is an average of the reals in that interval. Set theory contradicts this area of mathematics, which means it isn't the foundation for all math. > >>>>> It seem your argument is based on the idea that infinites do not exist >>>>> in physical reality. But mathematics is abstract, so this seems an >>>>> absurd objection. >>>> I think if Wolfgang and Han were offered a more sensible treatment of >>>> the infinite case, they might find it more palatable. >>> Uh, like yours you mean? Snicker. Han rejects completely the existence >>> of a "completed infinity". This is pretty close to the opposite of what >>> you're trying to do. >> That is true, for the reason I just mentioned. The standard streatment >> is nonsensical. > > You really do have a hugely inflated opinion of yourself. That's a survival technique in Kroneckerland. > The standard treatment makes a huge amount of sense. Define "huge" and "sense". > That you are unable to grasp it points to a problem with you, not with the treatment. That I am able to grasp it enough for you to leave questions unanswered bodes ill with you. That areas of mathematics contradict the "foundations" thereof is unacceptable. > If you > additionally think that anyone but you is ever going to find sense in > your treatment, I'd call you delusional. You can call me Ted if that makes you feel better. There are plenty out there that get bits and pieces of what I'm saying. Your discouragement is to be expected. :) > >>>>> If you refuse the idea of infinite sets, what does it mean to you to >>>>> say a function has domain and range R? >>>>> >>>> What does it mean to you, if not that one can use that range as a >>>> constant for infinite sets? Why can't we say that, over the entire range >>>> of R, the naturals have twice the density of the evens, and so are twice >>>> as large a set? >>> You're free to do so. The evens do have have a density of 1/2 in the >>> naturals (not in R, I'm not sure you meant to say R. I think they both >>> have 0 density in R...). If you want to think of this as meaning the >>> set is twice the "size" then you're free to do so. >> I said "the range of R", meaning over the real line. > > Well, this is your own terminology that really doesn't mean anything > much to me. "The number line" is a visual aid used in kindergarten, not > a central part of formal mathematics. You've certainly never defined > it. I believe it was Virgil (could be wrong) who used the term, today or yesterday, "over the range of R". Maybe it was Stephen. In any case, it's obvious what it means. But so, you agree that, given certain considerations, the set of even naturals can be said to be half the size of the set of naturals? If so, then aren't the conclusions of cardinality not generally true, if equivalent cardinality is
From: Virgil on 17 Sep 2006 20:03
In article <450d6d1e(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>> Mike Kelly wrote: > >>> > >>>> Given that any second-year student of probability theory knows that > >>>> there are no uniform distributions over countable sample spaces, [ ... ] > >>> This "given" is most disturbing. Mainstream mathematics is so certain > >>> about its own right that no sensible debate is possible. > >> There IS no LUB on the finites, omega notwithstanding. Omega's a > >> phantom. That's why you can't get any average value or any uniform > >> probability distribution. > > > > Vaguely correct, minus reflexive whining about Omega. > > > >> In general, it doesn't make sense to talk > >> about probability without a uniform probability distribution over a > >> finite set. > > > > That's absurd. I don't think you meant to say what you said here. Of > > course there are non-uniform probability distributions and probability > > distributions on infinite sets. > > Okay. I misspoke. But what about uniform probability distributions on > infinite sets in general? The answers to such questions are part of the content of measure theory, q.v. > > > > >> However, since probability is really a percentage, > > > > That is, a real number between 0 and 1. > > Yes. > > > > >> any > >> subset which is a finite fraction of the whole can certainly have a > >> probability associated with it: that fraction. > > > > Only if a uniform distribution can be defined on the whole. > > Why? A probability IS a fraction. Not necessarily. It is always a real between 0 and 1 inclusive, but not all reals in that range are fractions. |