An uncountable countable set
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From: Tony Orlow on 20 Aug 2006 10:27 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}. TO
From: Virgil on 20 Aug 2006 10:29 In article <ec9o8p$4ne$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > In article <ec8hlf$qq8$1(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > > > > > >> 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? > > > > Only if (aleph_0^2+aleph_0)/2 = aleph_0. > > Which it does, in standard theory. It is not until you get to 2^aleph_0 > that you are able to distinguish aleph_0 from a larger infinity. > > > > > But even then, one must have some prior definition of addition of > > infinite cardinalities, and some prior definition multiplication of > > infinite cardinalities (or at least squarings of them), and some prior > > definition of division infinite cardinalities by finite cardinalities > > (or at least halving of infinite cardinalities), none of which TO has > > provided, so his "formula" is all nonsense. > > > > We have the prior definitions of all those operations with regards to > variables in general. On the contrary, each operation has been defined only for particular domains, none of which domains include any infinite quantities. In fact, what we call a single operation is often several operations. For example addition has one definition for naturals, a different definition for rationals, yet a different one for reals and another different one for complex numbers. So definitions of operations are valid only for their original domains of definition and new definitions are required for different domains of definition. Anyone wishing to extend an operation only defined on one domain to a larger domain must give a clear definition of what that extended definition is to mean. TO has not done this. > All that's additionally required is allowing the > variables to assume infinite values. > > > > Only a nutcase would suppose that an infinite cardinal need behave > > entirely like a finite cardinal anyway. > > Transfinite cardinals and limit ordinals don't behave like numbers or > quantities at all, but it's not nutty to expect numbers to act like > numbers. it is no more nutty to expect infinite ordinals or cardinals to behave differently from finite ordinals or cardinals that to expect even naturals to behave differently from odd naturals. > It's nutty to allow them to act like potions and incantations. Then TO is confessing his nuttiness, as that is precisely what he is doing. > > Yeah, that would the the lst natural. We all know THAT doesn't exist. > Or, is it aleph_0? Same thing anyway. ;) And here I though the first natural was 0 or 1, depending on whose model one was following.
From: Tony Orlow on 20 Aug 2006 10:37 mueckenh(a)rz.fh-augsburg.de wrote: > Tony Orlow schrieb: > >> Hi WM. I agree entirely with your analysis regarding the interdependence >> of the existence of the infinite set of naturals and the existence of >> naturals of infinite value. This leads to the conclusion that the set of >> finite naturals is finite, though unbounded, since as you say, there >> cannot be an infinite number of differences of 1 where no two elements >> are infinitely different. In order for an infinite difference to occur >> between two natural, one must have an infinite value. > > Hi TO. This opinion is easily proved by Franziska's recognition that > there are omega differences. It is simple to see that omega*1 = omega. Yes, well, I'm not sure she's convinced yet. ;) > >> However, I wonder whether you would ever consider the existence of >> infinite natural numbers. > > Then these 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 would you call them? I suppose they should be called hypernaturals, to distinguish from the more limited standard naturals. Would you consider the existence of any infinite values? > >> After all, your argument says that either you >> have a finite set OR you have infinite values in the set. Is the second >> option objectionable for you? > > What would infinite values be good for? Distinguishable infinite values > are provably inconsistent (see my binary tree) and to have one infinity > the approved potential infinity is sufficient. Sufficient for what? How do you address the set of points in the continuum? Is that not an actually infinite set? I'm not sure what inconsistency you showed with your tree, but since you are addressing transfinite set theory, I'll assume that the source of the problem. I have problems with Virgil leafless tree and uncountable paths as well. I am talking about something outside of transfinite set theory - infinite values based on an infinite unit, formulaically comparable by extending induction to include non-finite n e *N. They are very good for addressing the Continuum Hypothesis (it actually makes it a non-question) and integrating true spatial measure with set theory. Perhaps you will consider it in time. I hope so. > > Regards, WM > :) TO
From: David R Tribble on 20 Aug 2006 10:44 Tony Orlow wrote: David R Tribble wrote: mueckenh wrote: >> Learn: Infinitely many stairs require infinite height. But that it not >> admitted in mathematics. There have to be infinitely many stairs which >> give an infinitely long staircase ... > David R Tribble wrote: >> Yes, infinitely many finite-sized steps produce a staircase of infinite >> height. That is admitted in mathematics. > Tony Orlow wrote: > All too often "admitted" that those infinities are equal. Yes, mathematicians accept the proofs that countably infinite sets have the same cardinality. mueckenh wrote: >> ... but this staircase is allegedly of finite height. > David R Tribble wrote: >> No, the staircase is infinite in height, but each step is finite in >> height. The distance between any two steps is finite, but the total >> height of all the steps together (the entire staircase) is infinite. > Tony Orlow wrote: > So, the greatest possible value that any natural number can reach, in > height, is infinite? You're almost right. The least upper bound of the naturals (and the reals, for that matter) is infinite (omega). Which is to say that naturals (and reals) can be as large as you like, but they never actually "reach" their LUB value. Thus they are all finite.
From: Tony Orlow on 20 Aug 2006 10:44
mueckenh(a)rz.fh-augsburg.de wrote: > Tony Orlow schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: >>> Franziska Neugebauer schrieb: >>> >>>> You have agreed to that this limit does not exist ("There is no L in >>>> N"). So the sum is at least _not_ _finite_ (not in omega). >>>> >>> There is no L, nowhere. It is wrong to say that L is in |N and it is >>> wrong to say that L is larger than any n e |N. Actual infinity does not >>> exist! I told you that already several times. >>> But refuse to understand what "to exist" means. In no case infinitely >>> many differeces of 1 can exist unless infinitely many diferences of 1 >>> do exist. But that means an infinite size. >>> >>> Regards, WM. >>> >> Well, here you answer my question just posed to you in my last post. >> There is no infinite number, in your opinion. But, how do you reconcile >> this with the infinity of reals in any unit interval? If, between any >> two distinguishable reals there is a third distinguishable from the >> first two, then does this not imply that we can subdivide the unit >> interval infinitely, yielding an infinite set of interval endpoints, aka >> reals? Isn't the set of reals in (0,1] infinite, and how do you >> characterize this number? > > Their number is potentially infinite, i. e., you can "create" > (Dedekind) and use as many as you want (but not more than 10^100 > numbers because of practical reasons). > > Regards, WM > Um, I would say the number of representations you can generate, or the number of points you can identify, given a finite system of subdivision, is potentially infinite, in a sense. But, no finite number of such iterations will cover all points in the interval, so one requires more than any finite number to complete this process. Therefore, there are an actually infinite number of points in the interval, even if any process of identifying them is only potentially infinite. Practical reasons aside, even at 10^100 subdivisions, you still have a finite interval, which can be further subdivided. Do you see the distinction between the generation of the points using a finite system, and their prior existence independent of any finite system of enumeration? There are actually an infinity of points in there. :) TO |