An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Mike Kelly on 3 Nov 2006 16:51 Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> stephen(a)nomail.com wrote: > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>> stephen(a)nomail.com wrote: > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>> stephen(a)nomail.com wrote: > >>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>>>> stephen(a)nomail.com wrote: > >>>>>>> <snip> > >>>>>>> > >>>>>>>>> What does that have to do with the sets IN and OUT? IN and OUT are > >>>>>>>>> the same set. You claimed I was losing the "formulaic relationship" > >>>>>>>>> between the sets. So I still do not know what you meant by that > >>>>>>>>> statement. Once again > >>>>>>>>> IN = { n | -1/(2^(floor(n/10))) < 0 } > >>>>>>>>> OUT = { n | -1/(2^n) < 0 } > >>>>>>>>> > >>>>>>>> I mean the formula relating the number In to the number OUT for any n. > >>>>>>>> That is given by out(in) = in/10. > >>>>>>> What number IN? There is one set named IN, and one set named OUT. > >>>>>>> There is no number IN. I have no idea what you think out(in) is > >>>>>>> supposed to be. OUT and IN are sets, not functions. > >>>>>>> > >>>>>> OH. So, sets don't have sizes which are numbers, at least at particular > >>>>>> moments. I see.... > >>>>> If that is what you meant, then you should have said that. > >>>>> And technically speaking, sets do not have sizes which are numbers, > >>>>> unless by "size" you mean cardinality, and by "number" you include > >>>>> transfinite cardinals. > >>>> So, cardinality is the only definition of set size which you will > >>>> consider.....your loss. > >>> If somebody presents another definition of set size, I will > >>> consider it. You have not presented such a definition. > >>> > >>> > >> I have presented an approach that works for the majority of infinite > >> bijections, and explained some of the exceptions. IFR works for all > >> numeric sets mapped from a common set. N=S^L works for all languages, > >> including those that express the first set. Both work on a parameteric > >> basis, using infinite case induction to finely order the values of > >> formulas for a specific infinite n. Rare exceptions include the set 1/n > >> for neN, whose inverse is itself, which IFR ends up saying has size 1, > >> but that's because the natural indexes and fractional mapped reals only > >> share one point in their range, 1. So, I think Bigulosity is worth > >> considering. > > > > Why? What is it good for? What theories is it used in? > > > > Bigulosity Theory. So, all Bigulosity theory does is tell you "The Bigulosity of set so and so is such and such"? How is that in any way useful? -- mike.
From: David Marcus on 3 Nov 2006 17:13 Mike Kelly wrote: > Tony Orlow wrote: > > Mike Kelly wrote: > > > Tony Orlow wrote: > > >> Mike Kelly wrote: > > >>> Tony Orlow wrote: > > >>>> Mike Kelly wrote: > > >>> <snip> > > >>>>> Now correct me if I'm wrong, but I think you agreed that every > > >>>>> "specific" ball has been removed before noon. And indeed the problem > > >>>>> statement doesn't mention any "non-specific" balls, so it seems that > > >>>>> the vase must be empty. However, you believe that in order to "reach > > >>>>> noon" one must have iterations where "non specific" balls without > > >>>>> natural numbers are inserted into the vase and thus, if the problem > > >>>>> makes sense and "noon" is meaningful, the vase is non-empty at noon. Is > > >>>>> this a fair summary of your position? > > >>>>> > > >>>>> If so, I'd like to make clear that I have no idea in the world why you > > >>>>> hold such a notion. It seems utterly illogical to me and it baffles me > > >>>>> why you hold to it so doggedly. So, I'd like to try and understand why > > >>>>> you think that it is the case. If you can explain it cogently, maybe > > >>>>> I'll be convinced that you make sense. And maybe if you can't explain, > > >>>>> you'll admit that you might be wrong? > > >>>>> > > >>>>> Let's start simply so there is less room for mutual incomprehension. > > >>>>> Let's imagine a new experiment. In this experiment, we have the same > > >>>>> infinite vase and the same infinite set of balls with natural numbers > > >>>>> on them. Let's call the time one minute to noon -1 and noon 0. Note > > >>>>> that time is a real-valued variable that can have any real value. At > > >>>>> time -1/n we insert ball n into the vase. > > >>>>> > > >>>>> My question : what do you think is in the vase at noon? > > >>>>> > > >>>> A countable infinity of balls. > > >>> 1) It's not clear to me what you mean by that phrase but I'll assume > > >>> the standard definition. Still, the question remains of which balls you > > >>> think are in the vase? Does every natural number, n, have a ball in the > > >>> vase labelled with that n? > > >> Conceptually, sure. > > > > > > Yes or no? What is the set of balls in the vase at noon? Which balls > > > are in the vase and which are not? > > > > > >>> 2) How come noon "exists" in this experiment but it didn't exist in the > > >>> original experiment? Or did you give up on claiming noon doesn't > > >>> "exist"? What does that mean, anyway? > > >> Nothing is allowed to happen at noon in either experiment. > > > > > > Nothing "happens" at noon? I take this to mean that there is no > > > insertion or removal of balls at noon, yes? Well, I agree with that. > > > But what relevence does this have to the statement "noon does not > > > exist"? What does that even *mean*? > > > > > > When you've been saying "noon doesn't exist", you actually mean to say > > > "no insertion or removal of balls occurs at noon"? > > > > > > How about this experiment, does noon "exist" in this experiment : > > > > > > Insert a ball labelled "1" into the vase at one minute to noon. > > > > > > ? > > > > > >> They both end up with countably many balls in the vase at noon. > > > > > > For now, I am going to try to restrict myself to discussing this new > > > experiment, because I want to understand what "noon doesn't exist" is > > > supposed to mean. And, again, your answer is ambiguous. I asked which > > > balls are in the vase at noon, not the cardinality of the set of balls > > > in the vase at noon. I then asked whether "noon exists", not whether > > > anything "happens" at noon. Please try answering the questions people > > > actually ask; it aids in communication. > > Why have you ignored everything I said above here? Are you simply > unwilling to critically examine the assumptions and inferences you > make? Maybe you just missed that part of my post, I'll ask again. > > What is the set of balls in the vase at noon in the simplified > experiment? > What does "noon does not exist" mean? Maybe Tony has given up on this discussion. -- David Marcus
From: Lester Zick on 3 Nov 2006 17:49 Well, since no one chooses to reply or even acknoweldge the following I think I'll add a few remarks myself. On Wed, 01 Nov 2006 12:50:10 -0700, Lester Zick <dontbother(a)nowhere.net> wrote: >On Tue, 31 Oct 2006 10:30:08 -0500, Tony Orlow <tony(a)lightlink.com> >wrote: > >[. . .] > >(Tony, since we have an already established audience on this thread >I'm piggybacking what amounts to a new line of reasoning to your post >to which I've already replied in detail. I hope the following explains >exactly the origins and significance of mathematical and arithmetic >infinities in purely mechanical terms.) > > > Real Theory > ~v~~ > >I'm now of the opinion that there is a specific reason why modern math >and set analysis are wrong in fundamental mechanical terms. The >difficulty has to do with what are thought of as real number lines and >their supposed characteristics as lines. In effect if we use the Peano >axioms and the suc( ) axiom to generate the naturals, we lock into a >system of straight line segments which never correspond to curves and >transcendental numbers and infinities drawn in terms of those curves. > >In other words it is quite possible to generate straight lines in such >terms but there can never be an exact equivalence between those lines >and any kind of transcendental infinity. Thus we can treat arithmetic >infinites which exist in terms of infinitesimal subdivision of >straight lines such as irrationals like the square root of two but not >those which exist in terms of transcendental infinities such as pi. > >On the other hand, however, we can proceed in the opposite direction >quite easily by generating straight lines as tangents through Newton's >calculus and his method of tangents. And in so doing we can develop >all possible reals through the mere assumption of curves instead of >straight line segments and infinitesimal subdivision instead of Peano >axioms and the suc( ) axiom. > >The problem is and always has been that mathematics in general is not >arithmetic in particular. And we can always generate straight lines in >terms of curves through tangency but not vice versa because in terms >of form there is only one straight line but infinite kinds of curves >to which there are straight line tangents and we can't proceed from >any straight line tangent backwards to any specific curve. > >In effect then arithmetic theoreticians have straight lines but they >cannot deduce curves from those straight lines and are forced to >imagine such collateral forms of infinity either superimposed on >straight lines themselves, such as imaginary real number lines, or in >some other group altogether. But in neither case can they deduce the >existence and properties of such transcendental infinities from the >existence and properties of straight lines produced by arithmetic >axioms except through approximation with straight line segments. > > > Mechanical Implications > ~v~~ > >Although our primary interest here is mathematical there is a good >deal more significance than mere conventional mathematics would >suggest or imply. We and all forms of being operate and think in terms >of curves or at least in non straight line forms through tautological >negation which is demonstrably true. However at the conceptual level >we communicate with one another through straight linear forms. This is >only true and possible because there is only the one form of straight >line but an unlimited number of curves. > >Thus each of us operating at the level of abstract thought has to >reduce curvilinear tautological results to straight line tangents in >order to compare to and communicate thoughts of one ontological >individual with those of another. And this process is exact through >tangency with those curvilinear tautological results. However then we >are left to ponder the origin of those exact results because we cannot >reverse the process in exhaustive mechanical terms to determine with >which curvilinear tautological form the tangency originated. > >~v~~ In mathematics it is often assumed that one can do whatever one chooses through the definition of axiomatic assumptions. Possibly the reasonableness of this idea can be established historically by earliest forms of geometric analysis where the existence of points, straight lines, and curves were simply taken for granted because commensuration of such elements seemed a more urgent problem. And yet one of the primary functions of mathematics is the integration of all conceptual realms and not merely a wishing away of differences through facile assumptions. Apparently we have no choice but to infer the existence of curves in various forms. But we have no corresponding need to infer the existence of straight lines as well because these we can derive through Newton's method of drawing tangents to curves. In other word one cannot integrate what one has not disintegrated. In mathematical parlance this means one cannot just willy-nilly integrate what has not first been differentiated because we have no derivative to indicate direction, no limits to indicate boundaries, and no functions to indicate variations. We can assume all these things but when we do there is no limit to our imagination and we risk giving the impression that precedence is the other way around, that somehow we can proceed from straight lines to curves when causation is actually in the opposite direction. Of course for purposes of commensuration the residuum is the straight line and its subdivisions. But the ancestry and lineage in tangency of straight lines must never be forgotten and we must always recognize transcendental curves as mathematically primitive to straight lines. ~v~~
From: MoeBlee on 3 Nov 2006 17:49 Lester Zick wrote: > On 2 Nov 2006 15:26:22 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > >Lester Zick wrote: > >> On 2 Nov 2006 11:21:10 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > >> > >> >Lester Zick wrote: > >> >> I'm perfectly content to stay right here while I refine my definitions > >> >> and terminology to bring them into better conformance with standard > >> >> mathematical usage and more appropriate neomathematical forensic > >> >> modalities. > >> > >> Evasion noted. > > > >You just quoted yourself, then commented 'Evasion noted'. Nicely done. > > Oh well considering the context you dropped somehow I'm not surprized. I looked over the context three times. If I excluded anything that has a material bearing on what I did include and my remark on it, then you can mention those exclusions specifically. Posts are not discredited by the mere fact that they don't include all previous quoted matter. Nobody has an obligation to include all previous quoted matter in every one of his or her posts. MoeBlee
From: Mike Kelly on 3 Nov 2006 18:44
Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> stephen(a)nomail.com wrote: > >>> So if instead, someone had just posed this problem > >>> Let > >>> IN = { n | -1/2^(floor(n/10)) < 0 } > >>> OUT = { n | -1/2^n < 0 } > >>> > >>> What is | IN - OUT | there would be controversy. > >>> Note, there are not balls or vases, or times or anything > >>> in this problem. Just two sets. It would help if you > >>> bother to stop and think for a second before responding. > >>> > >> Where are the iterations mentioned there? You're missing the crucial > >> part of the experiment. By your logic, you could put them in in any > >> order and remove them in any order, and when you say both processes are > >> done, nothing's left, but that's BS. It ignores the sequence specified. > >> This is just a distraction. > > > > Yes, if you insert and remove exactly the same balls then you get the > > same result when you're done, no matter what order you did it all in. > > Why is that BS? It seems blindingly obvious. > > It's BS applied to the problem at hand, because sets entirely ignore the > correspondence between insertions and removals over t or n, and try to > use omega or aleph_0 as some actual number, when the Twilight Zone > between the finite and the infinite, the largest finite or smallest > infinite, is exactly like the Twilight Zone between the smallest > positive real and 0. There is no smallest possible infinite number, or > it would have a finite number finitely before it. You are burying the > correspondence between t, n, and f(n) in the time vortex. It's a hat > trick. Next.... Mmm... babble. > > But I forgot, you think that if you shift all the insertions 1 minute > > further back in time, you DO get an empty vase at noon, right? I really > > don't understand how your mind works. > > > > You don't understand the contingency between a *removal of 1 and a > subsequent *addition of 10 before any further *removals of 1? Where is > that covered in your set-theoretic schema? I see no t in there, and only > vague reference to n. No, I don't understand. Why does it matter what order the balls are inserted and removed, if each and every one is inserted and later removed precisely once before noon? -- mike. |