An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Tony Orlow on 20 Aug 2006 22:37 Dik T. Winter wrote: > In article <ec9rfa$7tb$2(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > Dik T. Winter wrote: > > > In article <ec8hsd$r5t$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > > > Dik T. Winter wrote: > > > > > In article <ec7gek$frg$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > > > > > Inductively provable: Any set of consecutive naturals starting > > > > > > at 1 always has its largest element equal to size of the set. > > > > > > > > > > What is the size of the empty set? What is the largest element of the > > > > > empty set? > > > > > > > > The empty set is not a set of consecutive naturals starting at 1, > > > > because it does not include 1 as an element, and therefore it does not > > > > stand as a counterexample. However, even in your von Neumann ordinals, > > > > the empty set is 0, and contains zero elements, and could be considered > > > > the 0th in this sequence of sets. > > > > > > 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}, ... > > If by convention one says the empty set has max=0, then the empty set > > can be included this way. Otherwise, just leave it out and start with {1}. > > But you make conclusions about an ordinal number. I make conclusions about the ordinal number system, yes. > I question your "inductively provable". When you consider the von > Neumann ordinals I can state that the value of an ordinal is larger than > each of its elements. When you prove this inductively, how much larger is the size of the set than its maximum element? Is it always 1 larger than the max element? If so, then how does the set size at some point jump to infinity, while all elements remain finite? Is there a largest finite, and beyond it a point, beyond which everything is infinite? That's a very magical point, indeed! O, how I'd like to bring my unicorn there for a picnic! :D > I can use quite the same induction here as you are > using and come to the conclusion that the ordinal of the set of all naturals > is larger than each natural. With the vN ordinals, 1 greater, implying that aleph_0 is 1 greater than some finite. Nope. Won't work. You assume that what holds for natural numbers > also should hold for either ordinal or cardinal numbers. Finite cardinal > numbers can be equated in some sense to the non-negative integers, but they > are not identical. The same holds for finite ordinals. Ordinals and cardinals are like sepia prints. Sorry, was that harsh? I'll try to be more polite. I'm talking about scrapping ordinals and cardinals altogether, and sticking with fundamentals, like if P(x) for all x>y, and y's finite, then P applies not only to all finite x>y, but to positive infinite x as well. I'm talking about preserving the > relationship between proper superset and root set, and integrating measure into set theory entirely. > > Consider the following. The ordinal number of each initial segment of > ordinal numbers is larger than each of the ordinal numbers in that > segment. That works for finite and for infinite ordinals. Sure, but it's vague, and doesn't provide fine distinctions in size like my Inverse Function Rule. :) Tony
From: Tony Orlow on 20 Aug 2006 22:55 Dik T. Winter wrote: > In article <ec9r9u$7tb$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > Dik T. Winter wrote: > > > In article <ec8hlf$qq8$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > > > 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. 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. > > Yes, you can claim that, but you might get in trouble. We may similarly > state that if every term in a sequence A is greater than its corresponding > term in a sequence B, then the limit is obviously greater. Now apply that > to: lim{n -> oo} 1/n > lim{n -> oo} 1/(2n). You have to formalise what > you mean with an infinite sum of a diverging series before you can state > things like that. If, instead of talking about limits, we apply some unit infinity n, then we can maintain that 1/n>1/2n, one being half the infinitesimal value of the other. We might get in trouble, but I see none ahead, and no one's convinced me there's any down this stream. So, if we, for instance, say 2n<n^2 for n>2, then oo>2, and 2*oo<oo^2. Transfinitology says that 2*aleph_0=aleph_0^2. > > > claim there are infinitely many n e N, aleph_0 of them. So, if that > > formula represents the sum of the first x terms, and you plug in aleph_0 > > for x to include all of them, then you get that result. > > But that is no proof. No proof according to which set of assumptions? It's something of a proof of contradictions between sets of assumptions. > > > > > In > > > > standard theory, would this not equal aleph_0, and if so, does it make > > > > sense that sum(n=1->aleph_0: 1) = sum(n=1->aleph_0: n), when n>1 for > > > > all n>1? The scond would appear to be clearly a larger sum. > > > > > > No that does not make sense. sum{n = 1 .. oo} 1 is not defined, the same > > > holds for sum{n = 1 .. oo} n. If you want to use them you have to > > > provide a definition for them. > > > > All that needs doing is declaring a unit oo and allowing it to be used > > formulaically. sum{n = 1 .. x} 1=x, so sum{n = 1 .. oo} 1=oo. sum{n = 1 > > .. x} n=(x^2-x)/2, so sum{n = 1 .. oo} n=(oo^2-oo)/2. It was perhaps a > > year ago that three of us independently said the sum is (|N|^2-|N|)/2. > > It's not a problem dealing with infinite values, once you declare an > > infinite unit. "Diverges" doesn't specifically describe the value of the > > sum. :) > > Are you sure there will no contradictions come up? How do I operate with > oo? I am quite sure. I don't have a full spec at this time of arithmetic operations on Big'un (oo, unit infinity) and Lil'un (unit infinitesimal). Please ask specific questions. That would be helpful. > > > > He may be trying to point that out, but not in a way that I do understand. > > > Moreover, I have stated over and over again that aleph_0 does not behave > > > like a normal number. It is just people like WM and you that wish that > > > if behaves like a normal number, but it does not do so. I see no problem > > > with that. You can not find a midpoint in the ordered set of natural > > > numbers using a measure derived from the standard measure of the reals. > > > > True. WM and I and others want numbers to behave like numbers, and > > cardinalities and ordinals simply do not. > > Yes. The only problem you and WM have is with the terminology. But that > is just labels. You think it's just terminology? :| > > > We find them useless and a > > digression from the study of quantity and representation which we see as > > being the foundations of math. > > Yes, you find them useless. Others do not find them useless. Just opinion. Perhaps. The proof of the tree is the fruit, or the nut. :) > > > It irks people like us when set theory > > claims to be the foundation of math, and yet makes all sorts of > > exceptions and new rules for infinite values. > > That is not the case. The rules come from the way these things are > *defined*, not from anything else. Consider Conways surreal numbers, > they act much more in the way of numbers (they even form a field). Conway seems to have worked hard to make his system not conflict with standard set theory, and yet achieved a lot, in terms of making infinite values somewhat palatable. That's to be commended, but not necessarily emulated. > > > It's some kind of logical > > construction, but for folks like us, it's really not related to > > mathematics. > > It is related to mathematics. Perhaps it is not related to physics, > but it is related to computer science. Read about the computable > numbers as defined by Turing. None of Turing's work concentrated on the transfinite, that I've ever heard. I'd be very interested in even one quote of his on the topic. :) Tony
From: imaginatorium on 20 Aug 2006 23:00 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: > > imaginatorium(a)despammed.com wrote: > > > Tony Orlow wrote: > > >> David R Tribble wrote: > > >>> mueckenh wrote: <snip> I think this is the Outstanding Question: > > > Q1: "Is the greatest possible value that any natural number can reach > > > infinite in > > > height?" > > > > Okay, you can rephrase it like that, sure. It had a question mark anyway. Well, let's agree not to quibble about the slightly odd use of "height" here. Let's, equivalently say: Q1a: "Is the greatest possible value that any natural number can reach infinite in value?" What does it mean for a natural number to "reach"? Perhaps it means: let x (a variable) range over the entire set of natural numbers. If so, then plainly there is not a "greatest possible value". So do you mean: Q2: "Is the least upper bound of the natural numbers infinite?" The answer to this is simple: yes. Otherwise, you need to explain very carefully what else this means. Brian Chandler http://imaginatorium.org
From: Virgil on 20 Aug 2006 23:02 In article <ecavsb$qvf$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > TO claims a lot but does not deliver. > > > > I deliver but you leave the package outside the door. It is is too offal to bring inside. > > > >> 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. Not unless it can be given a numerical value. There are lots of infinite sums which can easily be distinguished from each other but all converge to the same value. That's easy > to do by extending induction to the infinite case and ordering formulas > on oo accoringly. Give us the formal axiom system in which this is allegedly valid and the formal statement of how TO's extended induction is alleged to work in that system so that we can verify or falsify it for ourselves. > > > > > > >> 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 TO wants them to behave like 3 and 3+1 and 3*2 and 2*3 and 3^3, but cannot show any reason they should, and there are convincing reasons why not. > > > > > 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. But |N| is not a number "on the real line" nor do any of the related expressions represent any real numbers. > > > > 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. At his present rate of learning about mathematics, TO will qualify for a G.E.D. and Social security at about the same time. 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. ;) Only to those well upwind. > > > > > > >> 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. ;) I already knew TO was mathematically inept, but now he shows himself to be socially inept too.
From: Virgil on 20 Aug 2006 23:04
In article <ecavtt$qvf$2(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > 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. ;) But even when wearing your best perfume you do not smell as nice. |