An uncountable countable set
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From: Virgil on 16 Sep 2006 18:47 In article <450c7444(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <450c6210(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > >> 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? Inclusive. > > Does 1/aleph_0 lie within the real interval [0,1]? AS "1/aleph_0" is not a real number at all, and real intervals contain nothing other than real numbers, "1/aleph_0" does not lie within ANY real interval.
From: Tony Orlow on 16 Sep 2006 19:21 Virgil wrote: > In article <450c7444(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <450c6210(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>> 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? > > Inclusive. Indeedly. And within what interval lies the multiplicative inverse of a number at least equal to 1? Is it [0,1]? >> Does 1/aleph_0 lie within the real interval [0,1]? > > AS "1/aleph_0" is not a real number at all, and real intervals contain > nothing other than real numbers, "1/aleph_0" does not lie within ANY > real interval. Can we not say that the multiplicative inverse of any real number greater than one is less than 1, and at least equal to 0?
From: Tony Orlow on 16 Sep 2006 19:25 Virgil wrote: > 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. > In the mathematical world, the greater framework can be considered relatively objective. > > >> 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. > Or conviction. > > > 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 19:32 Virgil wrote: > 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? Between which two naturals does Virgil find a difference of aleph_0-1? >> Is it fixed, or does it stretch and shrink at whim? > > The "length" of the real line does not exist at all. Then neither does the count of naturals, being the density of one element per unit difference times the number of unit differences, for the total count of naturals. >>> 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? I claim that, even if no length can be determined due to endlessness of a given line, that if that line exists, it is always as long as itself. Can you disagree? >>> 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. If you think the line is not measurable, then quit trying to pretend you are providing any kind of measure for sets which traverse the line. My logic is clear and simple. That you do not get it requires a lot of effort on your part, which is energy wasted. Tony
From: Virgil on 16 Sep 2006 20:29
In article <450c86f8(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <450c7444(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <450c6210(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > > > >>>> 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? > > > > Inclusive. > > Indeedly. And within what interval lies the multiplicative inverse of a > number at least equal to 1? Is it [0,1]? What "number" does TO refer to? Since Aleph_0 is not a real number, neither would it have a real number reciprocal. The set of non-zero real numbers is a group, so that only real numbers can have real number multiplicative inverses. > > >> Does 1/aleph_0 lie within the real interval [0,1]? > > > > AS "1/aleph_0" is not a real number at all, and real intervals contain > > nothing other than real numbers, "1/aleph_0" does not lie within ANY > > real interval. > > Can we not say that the multiplicative inverse of any real number > greater than one is less than 1, and at least equal to 0? It doesn't matter what TO tries to say about it, it still will not make either aleph_0 into a real number, nor any alleged reciprocal of aleph_0 into a real number. |