An uncountable countable set
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From: Tony Orlow on 16 Sep 2006 17:50 Aatu Koskensilta wrote: > David R Tribble wrote: >> The problem is that using too simple a language can lead to >> further confusion, or at least maintain the misunderstandings. >> I've been chastised for talking about "set size" instead of using the >> more specific term "cardinality", for instance. > > When discussing technical subjects there is no hope of avoiding > technicalities and technical concepts and terminology. But if someone > wonders about a feature of, say, a proof of Cantor's theorem it is > seldom useful, and often counterproductive, to bring in the heavy > machinery of first order logic and this or that formal set theory. Hi Aatu - I appreciate your desire to accommodate the simply naive and confused by addressing issues in terms they can understand regarding mathematical questions. I think that's very human and generous of you, and good advice to anyone trying to teach. I do think that once we get into foundational arguments of the sort going on here, we really can't avoid such technicalities, since the validity, if not the soundness, of the arguments hinges on such fundamental issues. I am sure you'd agree that one could hardly challenge the logical validity of such a theory without addressing the axioms and the logical operations allowable on them. Given the axioms and rules of inference, the conclusions are provably true or false. Soundness is another issue, regarding the fundamental justification for the logical axioms themselves, and whether they are "correct", meaning "objectively verifiable". That means that when we try adopting a set of axioms, we follow them to their conclusions, deductively. Then, we evaluate the axioms given the discrepancies between what they conclude and what we expected intuitively. If we find that our expectations are not met, then we must revise our theory and adjust our axioms or, even possibly, our rules of inference. A simple means of revising our axioms is to identify, intuitively at first and then more formally, what it is about the conclusions of the theory that violated our expectations, and create a rule that expresses that expectation as part of the axiom system. Then, we check through our axioms and find those that contradict the new rule. So, we can, inductively rather than deductively, home in on an appropriate axiom system, with the behaviors we'd like to preserve. > >> So there may be >> times when a more technically precise meaning is called for in >> order to cut through the confusion of multiple meanings. > > Of course, clarity is to be always sought, and in making clear exactly > what is meant by such ambiguous words as "size" in some context is often > necessary. > Yes, this is my point. When we speak of the "size" of a set, for finite sets it's the count. For infinite sets, some generalization becomes necessary. The issue for me is which tenets of finite sets do we consider most important to preserve as we generalize. The core problem is that adding an element does not make the "size" greater, much less by one. Neither does removing every other element make a set smaller, much less by half. When we even have a dense set on the real line like the rationals being equivalent to a sparse set like the naturals, then my sensibilities are especially offended. So, when I speak of size of a set, this means a measure which becomes larger when new elements are added, and smaller when existing elements are removed. If all math is based on sets with no sense of measure in the infinite case, then in the infinite case, there is no sense of measure. Is there any reason that you can see why such a system cannot be formulated, at least for some well defined sets? I think there are such systems, at least in some cases, such as measure theory and limiting density of a set, but do you see any clear obstacles to generalizing the measure of sets further so that it satisfies the most naive intuitions regarding infinite sets? Sequences are more than just sets. Thanks for your help, Sorry for blabbing :) Tony
From: Tony Orlow on 16 Sep 2006 18:01 Virgil wrote: > In article <450c6210(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <450bf8df(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>>>> 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. >>>>> >>>> Oooops, you're right. ANswering too many interrogatories at once. That >>>> should be aleph_0-n/aleph_0. Sorry 'bout that. >>> Still wrong, as aleph_0-n/aleph_0 = aleph_0-(n/aleph_0) = aleph_0, >>> if it means anything at all. >>> I suspect that TO meant to write "(aleph_0 - n) / aleph_0", which at >>> least makes a little sense. >> Yes, this time I left out the parentheses. Oops. Does it make " a little >> sense" now? > > Very little. Hi Virgil. :) It's only because "Things true in the finite case need not be true in the infinite case", or whatever you tend to say. I got sidetracked trying to look up an exact quote. :) I hope you're well. >>>>>>> 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. >>>> Take your vase and shake it a countably infinite number of times, and >>>> draw a ball. That should be random enough for you. >>> To chose something from a set "at random" has a technical meaning in >>> probability theory, meaning that each member of the set must have >>> exactly the same probability of be selected as any other member. >> Yes. For the finite naturals that would be 1/aleph_0, there being >> aleph_0 of them. > > But 1/aleph_0, not being a real number, cannot be a probability. >>> For a set with a countably infinite number of elements, such as the set >>> of naturals, that is not possible. >> Because the set does not have size aleph_0? > > Because there is no real number which can serve as a probability. > There is no real number r such that r + r + r + ... = 1. >>> But one can come close. Given any r strictly between 0 and 1, by >>> assigning the probability of (1-r)*r^n to each n in N = {0,1,2,3,...}, >>> one gets a legal probability distribution on N ( the sum of all such >>> probabilities equals 1) with the probability of consecutive naturals >>> being as nearly equal as one chooses, short of perfect equality, by >>> choosing r close to, but slightly less than, 1. >>> >>> Of course, if r were allowed to actually equal 1, all of those >>> probabilities become zero and no longer add up to anything except zero. >> That's an interesting concept. Let me cut and paste into notes...... >> >> Let's use 1/aleph_0 for r and see what happens. > > Since probabilities are necessarily real numbers and 1/aleph_0, whatever > it may be, is not a real number, it is also not a probability. Oh Papa San Grasshopper Grampa Boyeeee! Man! That's so cute. What is a probability? A real between 0 and 1? Does 1/aleph_0 lie within the real interval [0,1]? email me
From: Virgil on 16 Sep 2006 18:27 In article <450c63c7(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <450c3cfe(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> imaginatorium(a)despammed.com wrote: > >>> Tony Orlow wrote: > >>>> Randy Poe wrote: > >>>>> Tony Orlow wrote: > >>>>>> Randy Poe wrote: > >>>>>>> Tony Orlow wrote: > >>> <bibble-babble> > >>> > >>>> ... If the real line is considered a fixed range of aleph_0, > >>> Remind us what "range" means? Normally a range goes from a left end to > >>> a right end: in a set with a left end and no right end, where is the > >>> "range" measured to? > >> The range is not measured, but is declared constant over the real line, > >> whatever that length is. > > > > The real line does not have a length,. > > Length along a line can only be measured between points on that line and > > the real line does not have end points. > > Well, given the fact that there are n naturals in every real interval of > length n, I guess there is no measure of the number of naturals either. There is no measure of the "length" of the natural numbers. But that is quite a different matter. > > > > >>> A "fixed infinite length". Very funny (the first time, but has worn a > >>> bit thin by now)... > >> Snicker all you want. > > > > As all "lengths" are distances between points, what points does TO > > suggest one use to measure the "length" of the real line? > > We need not know the endpoints or length of a line to say it is as long > as itself. That is given. No, it is not "given" , because if a line does not have endpoints we do not even know that it has a length.
From: Virgil on 16 Sep 2006 18:32 In article <450c6449$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <450c3d37(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> imaginatorium(a)despammed.com wrote: > >>> Tony Orlow 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. You cannot say > >>>> that between any two naturals is another, in quantitative terms. I > >>>> meant, obviously, dense in the quantitative ordering. But, you knew that. > >>>> > >>>> So, that having been said, when there are an infinite number of > >>>> rationals for every half-open unit interval, and only one natural in > >>>> every such interval, how does it make sense that there are not > >>>> infinitely many more rationals than reals? Are the extra naturals that > >>>> make up the difference squashed down towards the infinite end of the > >>>> line, where there's no rationals left? Like I said, it's poppycock. > >>> > >>> Ah, the "infinite end of the line". What's that like, then? Sort of the > >>> end at the end that isn't there? > >>> > >>> Brian Chandler > >>> http://imaginatorium.org > >>> > >> In case you didn't note the sarcastic tone, that's me making fun of your > >> logic, not some part of my theory. Sheesh! > > > > TO keeps harping on the "length" of the real line as a part of his > > mythology. > > Does it exist? It's the number of naturals on the line. Between what two points does TO find that measure of length? > > Is it fixed, or does it stretch and shrink at whim? The "length" of the real line does not exist at all. > > > Length measurements on a line are distances on that line between two > > points on that line. > > Measurements of line SEGMENTS, yes. What other sorts of "length" measurements does TO claim he can make on a line? > > > Which points on the real line does TO use to measure the length of that > > line? > > It is not measurable, but like all objects and properties, is equal to > itself. If the line is not measurable then it does not have any length at all, just as TO has no sense at all.
From: Virgil on 16 Sep 2006 18:42
In article <450c71a1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Aatu Koskensilta wrote: > Given the axioms and rules of inference, the conclusions are provably > true or false. > > Soundness is another issue, regarding the fundamental justification for > the logical axioms themselves, and whether they are "correct", meaning > "objectively verifiable". If axioms were ever objectively verifiable they would not need to be assumed in the first place, but would be objectively verified. > That means that when we try adopting a set of > axioms, we follow them to their conclusions, deductively. Then, we > evaluate the axioms given the discrepancies between what they conclude > and what we expected intuitively. If we find that our expectations are > not met, then we must revise our theory and adjust our axioms or, even > possibly, our rules of inference. Sane people, as they grow up, recognize that they must occasionally revise their expectations. TO seems impervious to this reality of life. Which brings up the issue of TO's sanity. A simple means of revising our axioms > is to identify, intuitively at first and then more formally, what it is > about the conclusions of the theory that violated our expectations, and > create a rule that expresses that expectation as part of the axiom > system. Then, we check through our axioms and find those that contradict > the new rule. So, we can, inductively rather than deductively, home in > on an appropriate axiom system, with the behaviors we'd like to preserve. > > > > >> So there may be > >> times when a more technically precise meaning is called for in > >> order to cut through the confusion of multiple meanings. > > > > Of course, clarity is to be always sought, and in making clear exactly > > what is meant by such ambiguous words as "size" in some context is often > > necessary. > > > > Yes, this is my point. When we speak of the "size" of a set, for finite > sets it's the count. For infinite sets, some generalization becomes > necessary. The issue for me is which tenets of finite sets do we > consider most important to preserve as we generalize. The core problem > is that adding an element does not make the "size" greater, much less by > one. Neither does removing every other element make a set smaller, much > less by half. When we even have a dense set on the real line like the > rationals being equivalent to a sparse set like the naturals, then my > sensibilities are especially offended. So, when I speak of size of a > set, this means a measure which becomes larger when new elements are > added, and smaller when existing elements are removed. If all math is > based on sets with no sense of measure in the infinite case, then in the > infinite case, there is no sense of measure. > > Is there any reason that you can see why such a system cannot be > formulated, at least for some well defined sets? I think there are such > systems, at least in some cases, such as measure theory and limiting > density of a set, but do you see any clear obstacles to generalizing the > measure of sets further so that it satisfies the most naive intuitions > regarding infinite sets? > > Sequences are more than just sets. > > Thanks for your help, > > Sorry for blabbing :) > > Tony |