An uncountable countable set
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From: Virgil on 15 Sep 2006 23:06 In article <450b3f08$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Unbounded but finite may > >> be considered potentially, but not actually, infinite. > > > > That will be jiffy, once you give axioms and/or our definitions for > > 'potentially infinite' and 'actual infinite'. Until then, it's pure > > handwaving. > > > > MoeBlee > > > > As I said, a potentially infinite set is unbounded, but will all element > indices finite. It's "countable" in standard parlance. An actually > infinite set includes elements with infinite element indices, like 1/3 > has in the decimal reals. :) Is TO actually claiming that the decimal expansion of 1/3 has any digits with "infinite" indices? If so, he is farther from reality than anyone thought possible.
From: Virgil on 15 Sep 2006 23:22 In article <450b4261(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Randy Poe wrote: > >>> Tony Orlow wrote: > >>>> Haha!! Good one. "There are the same number of primes as there are > >>>> naturals, a proper superset." Good answer. > >>> Obviously you don't believe that. > >> Good guess. > >> > >>> Here's an equivalent statement: If n is a natural, > >>> there is an n-th prime. > >>> > >>> Do you think that's false? Do you think that there > >>> is a natural n such that there is an (n-1)th prime, > >>> but no n-th prime? > >> No, I don't think that, but I do recognize that the set of primes is a > >> subset of the naturals which proportion thereof has a limit of 0 as the > >> range of the naturals approaches oo. > >> > >> Do you honestly think bijection works as an infinite analog to equality? > > > > What does "as many as" if not "there's one of these > > for every one of those"? > > > > That's a long-winded way of saying: "yes". > > > > Besides, it follows by T-induction. The set of the first > > n naturals is the same size as the set of the first n > > primes for every n. Hence it's true for the whole sets. > > When you are comparing sets of points along the real line, which is what > primes and naturals and rationals and reals are, you notice that your > bijections stretch or shrink the measure between successive elements, > and so the value range is involved in the equation. It's like your > trying to change the size of a sealed balloon without changing the > temperature or pressure. The average number of elements per unit times > the range in units is the number of elements. More generally, if each > element is mapped from one of the rationals using a function f, the > inverse of that function over a given range gives the number of elements > in that range. If the real line is considered a fixed range of aleph_0, > then a set which is denser in every part of that range has more elements > than one which covers the same complete range with a lesser density. > That's the proper generalization from finite to infinite, and the > standard mistake to consider the real line to have whatever length > happens to be convenient, rather than a fixed infinite length for > purposes of comparison. > > > > >> Can a dense set like the rationals, with an infinite number of them > >> between any two naturals, really be no greater a set than the naturals, > >> which are an infinitesimal portion of the rationals? > > > > Yes. There's one of these for every one of those. > > Not per unit of value range. Within any unit interval (half-open) lies > exactly one natural, and an infinite number of rationals. > > > > >> That's just poppycock. > > > > Well, but that's a statement based on emotion rather > > than reason. > > No, I've made my reasons quite clear. > > > > > Reason says "there's one of these for every one > > of those". > > Only simple-minded reason which doesn't take into account the > relationship between average set density, value range, and element > count. What have they to do with "how many" there are. For finite sets, injection, surjection and bijection are the only valid tests of cardinality, so that naturally, TO will say that for infinite sets they cannot be. > The suggestion I am making is straightforward, the only > explanation I can see The fact that TO's "system" is not a system is why he is the only one who can see what does not exist. > for the refusal to consider it on the part of > "educated" mathematicians is an emotional response on their part That "emotional response" is merely the refusal to accept as true without a supporting system what we can prove false in any system as yet presented to us. Where is your axiom system. TO? Without it you have nothing.
From: Virgil on 15 Sep 2006 23:29 In article <450b4336$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > > > >> Can a dense set like the rationals, with an infinite number of them > >> between any two naturals, really be no greater a set than the naturals, > >> which are an infinitesimal portion of the rationals? That's just poppycock. > > > > A set is not dense onto itself. A set is dense under an ordering. And > > the set of natural numbers is dense under certain orderings. > > > > MoeBlee > > > > Not in the natural quantitative order on the real line. That particular ordering is irrelevant to the issue. Which of many possible order relations defined on a set one looks at is not a property of the set itself, but only of how one chooses to order it. > You cannot say > that between any two naturals is another, in quantitative terms. Why not? TO says much sillier things. There is no such thing as "in quantitative terms" without imposing an order relation on a set. And any ordering is as much an imposed ordering as any other.
From: Mike Kelly on 15 Sep 2006 23:49 Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Randy Poe wrote: > >>> Tony Orlow wrote: > >>>> Haha!! Good one. "There are the same number of primes as there are > >>>> naturals, a proper superset." Good answer. > >>> Obviously you don't believe that. > >> Good guess. > >> > >>> Here's an equivalent statement: If n is a natural, > >>> there is an n-th prime. > >>> > >>> Do you think that's false? Do you think that there > >>> is a natural n such that there is an (n-1)th prime, > >>> but no n-th prime? > >> No, I don't think that, but I do recognize that the set of primes is a > >> subset of the naturals which proportion thereof has a limit of 0 as the > >> range of the naturals approaches oo. > > > > So, uh, do you think that there is always an nth natural? And always an > > nth prime? So arne't there, like, the same number of both, given the > > equivalence between the two ideas? Why do you *always* avoid answering > > the most pertinent questions directly? > > I have answered that, directly, several times. No, I do not consider > simple bijection to be grounds for considering two sets equal, when the > value range is not determined. If you speak of all reals or all > naturals, that covers the same real range called "all", so if one set is > denser than another, and covers the same range, it's a bigger set. It's > that simple. If you claim the naturals and evens, for instance, have the > same count, that in essence says the evens have twice the range, since > every even is twice its corresponding natural. That's wrong. In essence, it says that you can count the naturals and evens off in pairs and never run out and also that eventually you reach any given one. Is this wrong? > >> Do you honestly think bijection works as an infinite analog to equality? > >> Can a dense set like the rationals, with an infinite number of them > >> between any two naturals, really be no greater a set than the naturals, > >> which are an infinitesimal portion of the rationals? > > > >> That's just poppycock. > > > > Doesn't matter if you feel that way. All cardinality says in set theory > > is "sets with this cardinality are bijectable with one another". > > Yes, it's the claim of mathematicians that this is a valid extension of > the finite meaning of "set size" which is so irksome. Irksome to you maybe. To most people it's a tool to guide intuition and communication. It offends your intuition? Fine. Just think of cardinality as meaning "bijectible" and nothing else. But this *in no way* affects the validity of the mathematics of standard set theory. Because mathematical theory doesn't say "cardinality is a good extension to the finite meaning of set size". That may be where the intuition that drove the original idea came from, it may be part of how it's explained to people, but it's not what cardinality actually *is*. Cardinality is a convenient way to point out classes of sets that are bijectible. That's *all* it is so objections of the form "but that's not what size is for infinite sets" are vacuous. >Go ahead and have > your broad equivalence classes based on nothing other than bijections, > but stop pretending that hocus pocus applies to anything outside of > Cantor's Garden O' Tricks, because it doesn't. You don't actually know any set theory other than a cursory knowledge of the axioms and natural language descriptions of some of the theorems, gleaned from the internet. You take great pride in knowing there is no point in studying it. So, you are unaware of what cardinality is for. You think the *point* of cardinality is to be able to proclaim "set X has size Z". Really, it's just a clever definition about sets that are bijectible to save an awful lot of words and ink. >Until the concept of > measure is applied to infinite sets of reals, they cannot be properly > compared. Who wants to "properly compare them"? Why? What is the *purpose* of doing this? To satisfy your intuition? What do your ideas actually add to mathematics other than the dubious claim that they make "more sense" to you? Are you aware that pretty much everything you've talked about to do with densities and "IFR" can probably be formulated in standard theory? And do you get why it's not a big mathematical topic? Because it isn't very interesting. It doesn't add much to what you can do. You're wasting your time creating an exruciatingly poor version of something that could be done in standard theory quite easily and is of limited value anyway. Tick. Tick. Tick. Tick. > > Your personal feelings on whether this really means they have "the same > > number" of elements seems highly irrelevant, given that all set theory > > "cares about" is that "these sets are bijectable", something which you > > don't seem to dispute (give or take one really stupid argument about > > the powerset of the naturals being countable). > > Sure, and set theory plays Banach-Tarski Ball, and Omega the Littlest > Giant, and The Elements that Didn't Matter, and other such nonsense. Nonsense because they offend your intuition? Because they describe things that can't be done in the real world? Because you don't understand? What? > > Most people intuitively think that if two sets can have all their > > elements paired up then they have "the same number of" elements. You > > don't. So what? Set theory doesn't "care" if that's what you think > > "bijectable" means or not. > > > > Some people do, and many are turned off by the hat tricks of > transfinitology. Sorry, I wasn't aware this was a discussion about how to encourage people to take an interest in mathematics. Is it? Quite a digression but we can go there if you want. Just don't pretend you're competent to criticse set theory on mathematical grounds. >It's not correct that you can cut a ball into five > pieces and reassemble it into two solid balls of the original size, Because you can't divide a physical ball to arbitary precision. > and > it's not true that you can put ten balls in a vase and take one out, > over and over, and ever get an empty vase. Because you can't physically perform an infinite sequence of operations in finite time. >If you don't think people > "care" whether what they think makes sense or not, then I guess you > don't care whether what you think makes sense. I wasn't discussing what people do or do not fin
From: Virgil on 15 Sep 2006 23:50
In article <450b4773(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Does the set of naturals with property P have a mapping function from > >> the naturals? :) > > > > Explain what it means for a property P to have a mapping function from > > the naturals xD > > There is a formula such that maps each unique natural to a unique > element of the set, such that no element is omitted. If you are talking > about a property that defines a subset of the naturals, it's bound to be > defined by some formula on the elements of N, no? How else do you intend > to define this set? > > > > > Pretend it does. Then explain how one determines the probability that a > > natural has property P. Go wild. For bonus credit explain what happens > > when property P doesn't have a mapping function from the naturals =D! > > Suppose it has the property that it is divisible by 3. That is a special case, and general rules cannot be demonstrated only by special cases, even when the special case is dealt with honestly. > So, we can map > the set of naturals satisfying that property using f(x)=3*x. The inverse > functions is....anyone? That's right, g(x)=x/3. Huh! So, over the entire > real line we could expect 1/3 of all naturals to satisfy this property Non sequitur. What is true for finite sets does not automaticaly carry over to infinite ones. > > Where P does not have a mapping, this technique cannot be applied. Do > you have a specific example, for bonus credit? Let P be the property that for given natural n, the decimal expansion of the square root of n has "more" even digits that odd digits. > > > > > Here are a few questions for you to practice your new theory on : > > > > I'll assume you mean the finite naturals, of size "aleph_0". If you mean > the hypernaturals through Big'un, replace as desired. > > > What is the probability that a number is greater than n? > (aleph_0-n)/n <1 (aleph_0 - n)/n will still be infinite, so TO is wrong to call it a probability at all, much less a probability less than 1. > > What is the probability that a number is greater than another number > > (also randomly chosen)? > 1/2 Given that any natural is "randomly chosen", which is an impossibility in the first place, the probability that another natural, equally randomly chosen, will be greater than the first is as close to 1 as makes no matter. > > What is the probability that a number is a perfect square? > sqrt(aleph_0) Probabilities are all between 0 and 1. So where between 0 and 1, inclusive, does TO's sqrt(aleph_0) fit? > > > What is the probability that the gcd((n^17)+9, ((n+1)17)+9) is not > > equal to 1 for "a number" n? > > I dunno. At least that is correct, but when TO says he doono, what he knows is frequently wrong, as in the above. > > > > > Don't forget I asked for a definition of "probability", too. I'm > > dreadfully ignorant on these matters o_O > > Probability requires a fixed range and sample space. Han's point is that > it's incompatible with transfinitology. It's not incompatible with all > notions of the infinite, though. That does not define probability. And while it is true that there is no uniform probability that can be defined on a countably infinite set, that does not mean that there cannot be any probability defined on such a set. > > > > Interesting. Both you and Han have implied that I don't know anything > > about probability. Based on what? That I think your ideas about > > mathematics are very, very stupid and will never amount to anything of > > any value whatsoever? Cute. > > > > Well, yes, that's part of it. Are you surprised? Since TO gives evidence of both ignorance of mathematics and stupidity about mathematics and a great deal more in every post, he is hardly in a position to carp. |