Cantor Confusion
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From: David Marcus on 12 Oct 2006 18:39 Virgil wrote: > In article <1160650371.242557.284430(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > In article <1160578706.221013.145300(a)c28g2000cwb.googlegroups.com> > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > So the definition I gave for a limit of a sequence of sets you agree > > > > > with? Or not? I am seriously confused. With the definition I gave, > > > > > lim{n = 1 .. oo} {n + 1, ..., 10n} = {}. > > > > > > > > Sorry, I don't understand your definition. > > > > > > What part of the definition do you not understand? I will repeat it here: > > > > What *might* be a sensible definition of a limit for a sequence of sets > > > > of > > > > naturals is, that (given each A_n is a set of naturals), the limit > > > > lim{n = 1 ... oo} A_n = A > > > > exists if and only if for every p in N, there is an n0, such that either > > > > (1) p in A_n for n > n0 > > > > or > > > > (2) p !in A_n for n > n0. > > > > In the first case p is in A, in the second case p !in A. > > > Pray, read the complete definition before you give comments. > > > > I do not believe that definition (2) is of any relevance. > > Cantor uses Lim{n} n = omega witout much ado. > > omega is simply defined as the limit of the increasing natural numbers. > > In his first paper he uses even Wallis' symbol oo. What should there > > require a definition, if all natural did exist? > > This is what I use and write in modern form: Lim {n-->oo} {1,2,3,..,n} > > = N. > > Where in ZFC or NBG does "Mueckenh"find any definition of any such limit? Or, in what book does mueckenh find this? > There are axioms and definitions in ZFC and NBG which allow for N, but > absent any statement of the axioms "Mueckenh" is allowing, "Mueckenh" > has no arguments at all. -- David Marcus
From: Tony Orlow on 12 Oct 2006 19:28 William Hughes wrote: > Tony Orlow wrote: >> William Hughes wrote: >>> Tony Orlow wrote: >>>> Dik T. Winter wrote: >>>>> In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> "Albrecht" <albstorz(a)gmx.de> writes: >>>>> > David Marcus schrieb: >>>>> ... >>>>> > > I don't follow. How do you know that the procedure that you gave >>>>> > > actually "defines/constructs" a natural number d? It seems that you keep >>>>> > > adding more and more digits to the number that you are constructing. >>>>> > >>>>> > What is the difference to the diagonal argument by Cantor? >>>>> >>>>> That a (to the right after a decimal point) infinite string of decimal >>>>> digits defines a real number, but that a (to the left) infinite string >>>>> of decimal digits does not define a natural number. >>>> It defines something. What do you call that? If the value up to and >>>> including every digit is finite, how can the string represetn anything >>>> but a finite value? >>>> >>> Because there are two types or strings. Strings that end and strings >>> that don't end. Only strings that end represent finite values. >>> >>> -William Hughes >>> >> And what about countably infinite strings which cannot achieve actually >> infinite values? > > If the strings do not end they do not represent finite values. Please prove this in specific mathematical terms, without using the circular argument involving the supposed infinity of omega. > > You have a hole between what most people call finite and > what you call infinite. Uh, no, you do. You argue that since the strings > cannot represent infinite things they must represent finite > things. This does not follow. So, they can represent a third kind of number, which is not finite, nor infinite? Pray tell, what sort be's such a number? You need at least three > catagories, finite, unbounded and infinite. Well, I have been lambasted for calling the countably infinite "finite but unbounded". Are those two mutualy exclusive? Is there an "unbounded" which is not "infinite? Please answer carefully, as I would not want you to make any mistakes. > Countably infinite strings have unbounded length and > do not represent finite values. > *Because* they are unbounded? Does "unbounded" MEAN *infinite*? > [ You have four types of ordered sets of integers. > > -sets that end with a finite integer > -sets that do not end but do not contain an infinite integer > -sets that end with an infinite integer > -sets that do not end and do contain an infinite integer > Do you actually allow these for types of sets of quantities? > You have never really come to terms with the second type of set. Those are the T-riffics and their family. > (You have never really come to terms with the fourth type of set, > but this type of set has rarely been discussed). Of course they have, but like unbounded sets of finites, they're rather hard to measure, especially given that one has to accept first of all the notion of a specific infinite quantity. > ] > > -William Hughes > > > I agree that > "countably infinite string which cannot actually achieve infinite > values" > are a problem with your theory. Your theory, your problem. > My take on reality, and not only mine, to which this thread, its origin, and contents, all attest.
From: Tony Orlow on 12 Oct 2006 19:31 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: >>> William Hughes schrieb: >>> >>>> mueckenh(a)rz.fh-augsburg.de wrote: >>>>> William Hughes schrieb: >>>>> >>>>> >>>>>>>> 0.1 >>>>>>>> 0.11 >>>>>>>> 0.111 >>>>>>>> ... >>>>>>>> >>>>>>> That is correct. But every element of the natural numbers is finite. >>>>>>> Hence every element covers its predecessors. If 0.111... is covered by >>>>>>> "the whole list", then it is covered by one element. That, however, is >>>>>>> excuded. >>>>>>> >>>>>> Since no one has claimed that '0.111... is covered by "the whole >>>>>> list"', I fail >>>>>> to see the relevence of a sentence that starts out >>>>>> 'If 0.111... is covered by "the whole list"'. >>>>> If every digit position is well defined, then 0.111... is covered "up >>>>> to every position" by the list numbers, which are simply the natural >>>>> indizes. I claim that covering "up to every" implies covering "every". >>>>> >>>> Quantifier dyslexia. >>> Quantifier magic may apply and may be useful at several occasions. But >>> to state that in a unary representation of natural numbers the union of >>> "up to every" and "every" have different meaning is easily disproved. >>> >>>> The fact that >>>> >>>> for every digit position N, there exists a natural number, M, >>>> such that M covers 0.111... to position N >>>> >>>> does not imply >>>> >>>> there exist a natural number M such that for every digit >>>> position N, M covers 0.111... to position N >>> In case of linear sets we have a third statement which is true without >>> doubt: >>> >>> 3) Every set of unary numbers which covers 0.111... to a finite >>> position N can be replaced by a single unary number. >>> This holds for every finite position N. If 0.111... has only finite >>> positions, then (3) holds for every position. As 0.111... does not >>> consist of mre than every position, it holds for the whole number >>> 0.111.... >>> >> So we have >> >> for every digit position N, there exists a set of >> unary numbers which covers 0.111... to position >> N >> >> and >> >> for every digit position N, if there exists a set of unary >> numbers which covers 0.111... to position N, >> there exists a single unary number, M, >> such that M covers 0.111... to position N >> >> this implies >> >> for every digit position N, >> there exists a single unary number, M, >> such that M covers 0.111... to position N >> >> >> this does not imply >> >> there exists a single unary number M such that for every digit >> position N, M covers 0.111... to position N > > Why shouldn't it? If every digit position of 0.111... is a finite > position then exactly this is implied. Your reluctance to accept it > shows only that you do not understand how an infinite set can consist > of finite numbers. In fact, nobody can understand it, because it is > impossible. > > Regards, WM > But Wolfgang, surely that consideration does not impact, say, the set of reals in (0,1], which are all finite, yet whose number is infinite. It is not a requirement that a set of all finite values be finite. That conclusion follows from the combination of that fact with the fact there is a constant positive unit difference between consecutive elements. TO
From: Tony Orlow on 12 Oct 2006 19:33 mueckenh(a)rz.fh-augsburg.de wrote: > briggs(a)encompasserve.org schrieb: > > >> Waving the magic wand, parameterizing by t and appealing to an intuitive >> notion of "if all steps of a process are defined it follows >> that the outcome is uniquely defined" is a poor substitute. That notion >> turns out to be false. > > That notion is not at all different from the notion pi = a_n*10(-n). > Only the consequences are clearer in this case. > >> If someone is willing to contemplate the one and not the other then >> you're not dealing with the mathematical part of the problem. You're >> dealing with delusions of physicality. The way out of that morass is >> not to scale things so that our intuitions are satisfied. It is to >> define things clearly enough that our implicit intuititive assumptions >> need not be silently invoked. > > There is no intuition involved if we find that lim {t-->oo} t > 1. This > can be obtained from lim {t-->oo} 1/t < 1. It is just pure mathematics. > > Regards, WM > Yes, based on the notion that x>1 <-> 1/x<1 and 0<=x<1 <-> 1/x >1. And then of course x=1 <-> 1/x=1.
From: David R Tribble on 12 Oct 2006 19:49
mueckenh wrote: >> Yes, but the assertion of Fraenkel and Levy was: "but if he lived >> forever then no part of his biography would remain unwritten". That is >> wrong, because the major part remains unwritten. > David R Tribble wrote: >> What part? > mueckenh wrote: >> That part accumulated to year t, i.e., 364*t. > David R Tribble schrieb: >> It's stated that he lives forever, so what value of t you are using? > mueckenh wrote: > You can use any positive value of t and prove that the unwritten part > n(t) for t > t_0 is larger than the unwritten part for t_0. You can > even use the formal convergence criterion for the convergent function > 1/n(t). There is no room for he assumption that the written part could > ever surpass the unwritten part. If I use any positive value for t, then there is still the positive value t+1 (and 2t, t^2, and all the rest), none of which satisfies the "lives forever" part. So I can't use any positive value of t. mueckenh wrote: >> If you think Lim {t-->oo} 364*t = 0, we need not continue to discuss. > David R Tribble schrieb: >> I don't think anyone has said that. I merely asked which pages (days) >> in the "major part" of the book don't get written. Do you have a >> certain t in mind? > mueckenh wrote: > I merely answer that it is completely irrelevant to speak of certain t. Then why did you say "use any positive value of t"? > The paradox is raised only by the asumption that the set of all t did > exist. What paradox? |