An uncountable countable set
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From: Virgil on 20 Aug 2006 16:24 In article <eca5p5$n03$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > David R Tribble wrote: > > Dik T. Winter wrote: > >>> Let L be the list of all natural numbers > >>> Let K be the sequence of digits such that for each p in L, K[p] = 1. > >>> What is *wrong* with that definition? > > > > Tony Orlow wrote: > >>> Sorry to interject, but could you tell me how long a sequence K is? If > >>> it's finite, then you only have a finite list L, and if it's countably > >>> infinite, then you have an uncountable list of naturals? > > > > Virgil wrote: > >>> How does having a countable list of naturals require an uncountable list > >>> of naturals? > > > > Tony Orlow wrote: > >> Is K a list of natural numbers? Pay attention. K is the sequence of > >> digits required to represent the naturals in binary. The set of binary > >> strings of length n has size 2^n. If n is aleph_0, then your countably > >> infinite string of digits produces uncountably many possible strings. > >> We've been through this. You cannot cull bits or strings in any way to > >> reduce this uncountably infinite set to produce the countably infinite > >> set of naturals you claim exists. In other words, there is no TYPE of > >> number which could represent the size of K. > > > > Yes, we've been though all this before and you still won't listen to > > reason. > > > > It takes ceil(log2(k)+1) binary digits to encode each natural k, which > > we'll call D(k). So: > > D(1) = 1, D(2) = 2, D(3) = 2, D(4) = 3, etc. > > > > Now add up all those D(k) for all natural k (all k in N), and call it > > t, the total number of binary digits for all naturals: > > t = sum D(k), for all k in N. > > Why would you sum them all, when each sequence of bit positions include > those before it? > > > > > Obviously, since each k is a finite natural, then each D(k) is a finite > > natural as well, i.e., each natural k requires a finite number of > > binary digits to represent it. Adding each finite D(k) to the finite > > sum D(1)+D(2)+...+D(k-1) up to that point yields a countable finite > > total number of digits for each k. > > Right, in your theory, which yields an uncountable number of binary > strings, ala 2^aleph_0. > > > > > There are a countably infinite number of naturals. Since the sum > > of a countable number of countable values is itself countable, t is > > countable. > > > > So, what is ceil(log2(aleph_0)+1) again? Undefined outside TOmania! > Is that countably infinite? It is unaccountably asinine. > You > have missed the point entirely, as usual. TO is too dull to have a point.
From: Tony Orlow on 20 Aug 2006 20:48 Virgil wrote: > In article <ec9r9u$7tb$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > >> Dik T. Winter wrote: >>> In article <ec8hlf$qq8$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: >>> > Dik T. Winter wrote: >>> ... >>> > > Eh? It holds for every finite number but does not hold for an >>> > > infinite >>> > > number of finite numbers? What do you mean with that? If there are >>> > > infinitely many finite numbers, I would say that as it holds for >>> > > every finite number, it also holds for infinitely many finite numbers. >>> > >>> > If the formula applies to an infinite number of finites, then does the >>> > sum of the aleph_0 finite naturals equal (aleph_0^2+aleph_0)/2? >>> >>> I think there is a misunderstanding. The formula >>> sum{i = 1 .. n} i = n * (n + 1) / 2 >>> holds for every finite number n, so it holds for infinitely many finite >>> numers n (as there are infinitely many finite numbers n). But we can >>> not switch to an infinite sum (that is something different). >> Well, we can. > > TO claims a lot but does not deliver. > I deliver but you leave the package outside the door. > >> If we turn to what we know about infinite series, we can >> apply notions such as, if every term in series A is greater than its >> corresponding term in series B, then the sum is obviously greater. > > Only if such a sum "exists". None of the standard ways of defining such > sums of infinitely many terms require arbitrary "sums" to have a value. > If such a sum is defined in such a way that it can be distinguished from other such sums, the it exists as far as being a construct. That's easy to do by extending induction to the infinite case and ordering formulas on oo accoringly. > > >> True. WM and I and others want numbers to behave like numbers > > TO wants numbers which are different from each other to behave as if > they were not different from each other(TO wants, in effect, even > numbers to behave like odd numbers and vice versa). Um, no, that's the opposite of what I want. I want |N| and |N|+1 and |N|*2 and |N|^2 and 2^|N| and |N|^|N| to all act as if there were different from each other, whereas in the standard theory those six are assigned two or three equivalence classes. I want finer distinction, not more obfuscation. > > Each type of number has its own properties, and requiring one type to > behave as if it were a different type, as TO keeps doing, is stupid. All that is on the real number line is the set of real numbers as points, and exactly one in every unit interval is a whole number. > >> and cardinalities and ordinals simply do not. > > > The finite ones behave like, and actually are, natural numbers. Not the non-finite ones. The finite ones are covered very well, thank you, by the ordinary naturals or integers. > >> We find them useless and a >> digression from the study of quantity and representation which we see as >> being the foundations of math. > > TO does not "find" mathematics at all, and TO needs several years of > concentrated mathematical study before he will be properly prepared to > take of the "foundations of math". > > Well, given my part-time endeavors over the last 2-3 decades and concerted efforts here and in reflection, it's coming along. The foundation begin with Leibniz and the definition of identity. The foundation of quantity is space, or geometry, and the foundation of logic is quantity. I'm cooking my approach. It's starting to smell pretty yummy. ;) > > >> It irks people like us when set theory >> claims to be the foundation of math > > It irks those of us who know a bit about mathematics that those who know > so little about mathematics as TO set themselves up as authorities on > what they know so little about. Oh. Sorry. I wasn't aware I was offending anybody. My apologies. ;) Tony
From: Tony Orlow on 20 Aug 2006 20:49 Virgil wrote: > In article <ec9rfa$7tb$2(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > >>> And the next in the sequence is {0} = 1, followed by {0, 1} = 2... >> That's the von Neumann ordinals. I am talking about {}, {1}, {1,2}, ... > > Which is no more that the von Neumann ordinals in drag. But I look so much better than John in purple velvet. ;) Tony
From: Tony Orlow on 20 Aug 2006 20:59 Virgil wrote: > In article <ec9s3c$8qv$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > >>> Then these infinite numbers would not deserve the name "natural number". >> Why is that? If they are whole numbers, each with successor, does that >> not fit the bill? > > What is your definition of "whole number", TO, that allows it to be > applied to objects not members of the minimal of the inductive sets > required by the axiom of infinity? A point on the real line exactly reachable with an endpoint by placing unit intervals end-to-end starting from 0 (using either a "countable" or "uncountable" number of intervals). > > One of TO's faults is that he will not allow any definition to mean > only what it states, but always has to suppose it to mean something else. > This is a very anti-mathematical anti-logical attitude, and explains > much of TO's difficulty in comprehending things logical and things > mathematical. > More vacuous dismissal. Whatever. > In ZF, ZFC andr NBG, a set, S, is called an inductive set if and only if > (1) {} is a member of S, and > (2) Whenever x is a member of S, then union {x,{x}} is a member of S. > The axioms of infinity in those systems declare that there exists > inductive sets, and other axioms guarantee that there must be a minimal > inductive set which is called the set of natural numbers and its > members, and only its members, are natural numbers in those systems. Very untrue. Those are the von Neumann ordinals, but that is not the general definition of an inductive set. It can start with any root set, and define successors in any manner it wishes. The digital number systems are inductive sets, ultimately, because each element may have more than one successor. In the most general sense, any tree is an inductive set, even if it's not at all regular, since only the root is not defined as successor to any other element. You really must expand your horizons, and take more vitamin C. ;) > > If TO wants to call anything else natural numbers, he has put himself > outside the pale. Into the tan, indeed. Out from the cave. Over the rainbow, if you will. Why not? Why object to infinite induction as an alternative to tranfinitology when it's no less reasonable? Why hide behind Georg? Boo!! TO
From: Tony Orlow on 20 Aug 2006 21:04
Virgil wrote: > In article <ec9ssu$9km$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > >> Virgil wrote: >>> In article <ec9nrs$44k$2(a)ruby.cit.cornell.edu>, >>> Tony Orlow <aeo6(a)cornell.edu> wrote: >>> >>>> Virgil wrote: >>>>> In article <ec8h1q$prm$1(a)ruby.cit.cornell.edu>, >>>>> Tony Orlow <aeo6(a)cornell.edu> wrote: >>>>> >>>>>> It would seem to me that any good theory >>>>>> of infinite sets could be applied to infinite sets of points, such as >>>>>> the reals in (0,1] or those in (0,2], and be able to draw conclusions >>>>>> such as that there is twice as many points and twice as much space in >>>>>> the second as in the first >>>>> That would require each point to occupy enough space that only a finite >>>>> number of them could fit into any finite interval. >>>> WRONG! >>>> >>>> It requires that some relationship be set up between a particular >>>> infinity of points and a finite length. What you suggest would be a >>>> finite set of finite elements, not any kind of infinite set. >>> If TO insists that infinitely many points can be compressed into a >>> finite space, then one can just as easily compress twice as many points >>> into the same space. >>> >>> If one takes the points of (0,2] and places each point from x at >>> position x/2 in (0,1], one has compressed "twice as many" points into >>> (0,1] as were originally there. Thus there cannot be any fixed >>> proportionality between the "number of points" in an interval and its >>> length for intervals of positive length. >> There is when one declares it. > > The problem being that one can declare as many different ones as one > chooses to declare, and each is just as valid as any other. Well, that's why one only decalres one to start with, and sees how that goes. Rather well, thanx. > >> Big'un is the number of reals in the unit >> interval, and the number of unit intervals on the infinite real line. > > I declare Card(P(N)) to be the number of reals in the unit interval and > Card(N) to be the number of unit intervals with disjoint interiors in > the infinite real line. Does this help you resolve the Continuum Hypothesis? That would be neat! :D > > TO's definition is either wrong (if his unit intervals are to have > disjoint interiors) or he has corrupted his own theory by letting the > umber of points in the unit interval equal the number of points in the > entire real line (if overlapping intervals are allowed, the number of > unit intervals in the real line equals the umber of points in the real > line). > As I told you, I entertain a much larger real line, with points actually infinitely many units from the origin, allowing for infinite reals, naturals and rationals (eventally even sedonions, prolly). > >> Where you claim to compress or stretch the inherent density of the real >> points on the line, you make proper measure impossible. > > Is TO going to pontificate on measure theory on the real line? What the hell else have I been doing? Oh, well, I guess I've made a few other points. But, yes, I advocate a means of melding measures. > > If so how is he going to deal with the inevitable non-measurable sets? Like the Cantor set, or what? I am up for a challenge, though to be honest, I think I need to write down what already cooking before I start a new pot. > > Cardinality has no problem with them. How does it deal with them? "They're all the same"? Sweet dreams, T |