Cantor Confusion
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From: Virgil on 12 Oct 2006 14:54 In article <1160648969.767158.122360(a)c28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1160578088.974689.303450(a)e3g2000cwe.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > William Hughes schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > [...] > > > > > > > > > There is no largest natural! There is a finite set of > > > > > arbitrarily large naturals. The size of the numbers is unbounded. > > > > > > > > > > > > > I can only conclude you have knocked youself out. > > > > > > Try the following gedanken-experiment to become accustomed with it: > > > a) How many different natural numbers can you store using a maximum of > > > 100 bits? > > > b) What is the largest natural number you can store with a maximum of > > > 100 bits? > > > > > > Regards, WM > > > > > > Answer to a) less than 100. > > > > I make the answer to A as 2^100, if each bit must be allowed to have its > > own personal bit position. > > And it must have its own personal value 0 or 1. Therefore you can > either represent one number with 100 bits or 100 numbers with one bit, > but the latter would not yield different numbers. Therefore the answer > is "less than 100". That there are different answers possible only reflects that the question was poorly posed.
From: Virgil on 12 Oct 2006 14:57 In article <1160649431.819176.107200(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David R Tribble schrieb: > > > Virgil schrieb: > > >> Note, the question originally asked was very careful to > > >> distinguish between the questions " Will the whole autobiography > > >> be written?", and "Will certain pages of the autobiography > > >> be written?, so my repharasing is accurate. > > > > > > > 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. > > > > It's stated that he lives forever, so what value of t you are using? > > > 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. No one has assumed that, unless it be "Mueckenh". > I merely answer that it is completely irrelevant to speak of certain t. > The paradox is raised only by the asumption that the set of all t did > exist. That causes no problems to anyone but "Mueckenh".
From: imaginatorium on 12 Oct 2006 15:01 Tony Orlow wrote: > Randy Poe wrote: > > Tony Orlow wrote: <snip> > >> 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 representations of finite values end, and the string doesn't > > end, so it breaks the rules of "strings that represent finite values". > > > > - Randy > > Can you rightly call it an infinite value? I can't. It's unbounded like > the finites themselves, but not infinite, as long as all digit positions > are finite. Yes, you (or rather, anyone _but_ you) can call it an "infinite value" if they have a clear definition of what "infinite" means, and it meets that definition. You can't really legitimately call it anything, since you have never defined what you mean by most words - the i-one in particular. Brian Chandler http://imaginatorium.org
From: Virgil on 12 Oct 2006 15:01 In article <1160649760.068206.172400(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1160578088.974689.303450(a)e3g2000cwe.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > William Hughes schrieb: > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > [...] > > > > > > > > > There is no largest natural! There is a finite set of > > > > > arbitrarily large naturals. The size of the numbers is unbounded. > > > > > > > > > > > > > I can only conclude you have knocked youself out. > > > > > > Try the following gedanken-experiment to become accustomed with it: > > > a) How many different natural numbers can you store using a maximum of > > > 100 bits? > > > b) What is the largest natural number you can store with a maximum of > > > 100 bits? > > > > What is the relevance? > > To inform the set theorist about the possible existence of sets with > finite cardinality but without a largest number. To which the set theorist quite rightly says, "not in my set theory". > Cantor assumed "the infinite set of finite numbers". That is impossile. There are axiom systems in which it is quite possible. And until "Mueckenh" shows that those systems are not only in his opinion paradoxical but contain internal contradictions that he can demonstrate to others, his fussing is futile.
From: Virgil on 12 Oct 2006 15:05
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? 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. |