Cantor Confusion
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From: Tony Orlow on 12 Oct 2006 11:05 Dik T. Winter wrote: > In article <452d140b(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > mueckenh(a)rz.fh-augsburg.de wrote: > ... > > I don't understand this definition either. You write lim{n=1 .. oo}. Is > > that supposed to be a sum over that range, > > Why should it be a sum? > > > or do you mean lim(n->oo)? > > What is the difference? > > > Does 1 belong there? > > Why not? It serves no purpose but to make the limit look like a sum over a range. Is lim{n=1 .. oo} any different from lim{n=100 .. oo} or lim{n=-1 .. oo}? Is that to specify that you are approaching oo from the left? Anyway.... > > > Also, you are looking for a limit of what, a set of > > balls from n+1 through 10n? Are you looking for the limit of the *size* > > of that set, which would be 10-(n+1)+1, or 9n? > > Pray re-read, I explicitly state that I am talking about a limit of sets. > > > If you want to put this > > in limit terms, with the corrections I suggest, you have the size of the > > set being lim(n->oo: 9n). That's not 0 by any stretch of the imagination. > > Pray re-read. I am *not* talking about the limit of the size of sets, but > about the limit of sets. And what are you saying about this set? That it cannot exist because n+1=10n for n=aleph_0? I hope not. Besides, is that even relevant to the problem?
From: Tony Orlow on 12 Oct 2006 11:09 Randy Poe 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. > > But not necessarily a number. > >> 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.
From: Tony Orlow on 12 Oct 2006 11:10 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?
From: Tony Orlow on 12 Oct 2006 11:14 Virgil wrote: > In article <452d14fe$1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> 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. > > An infinite string of digits. but every standard natural number is > defined by a finite string of digits, given a base, so those infinite > string define nothing at all. Besides themselves. > > > > What do you call that? > > An infinite string. > >> If the value up to and >> including every digit is finite, how can the string represetn anything >> but a finite value? > > If a binary string s:N --> {01} is such that s(n) = 1 for all n in N, > then its "value" is sum_{n in N} 2^n, which diverges. Of course it diverges, which means it attains an infinite value for infinite n. But, for all finite n, sum(x=0->n: 2^x) is finite. You have no infinite n in N. > > But all the partial sums, sum_{n =1..k}2^n are all finite. Right, and that's all there is in N. There is nothing in N that is infinite in value or in index. > > So the value up to and including every digit is finite and the string > itself cannot represent any finite value. It cannot represent any infinite value for that very reason. > > So another of TO's fairy tales is debunked. Dream on.
From: Tony Orlow on 12 Oct 2006 11:18
Dik T. Winter wrote: > In article <452d14fe$1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > Dik T. Winter wrote: > > > In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> "Albrecht" <albstorz(a)gmx.de> writes: > ... > > > > 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? > > I define it as a string of digits and it does not represent a number. It is > only when you give proper definitions of what strings extending infinitely > far away to the left represent, that you can talk about what it represents. > In common mathematics there is no such definition. When Peano defines the natural numbers, does he talk about what they represent, or only how they are generated? > > That infinite strings to the right define real numbers is entirely due to > the *definition* of real numbers. And that infinite strings to the left, > within the theory of p-adics, have specified meaning is entirely due to the > *definition* of p-adics. (And I may note that in the p-adics there is *no* > definition for infinite strings to the right.) Uh, isn't that what the p-adics define? Or, are you saying there is no quantity associated with any given p-adic, even though there is order and arithmetic within the system? > > In principle, infinite strings are just that. Within some theories you can > make consistent definitions for them, but that is all what it means. The > only current consistent theories I know (there may be more) is that 0.111... > as a decimal representation within the decimal numbers represents 1/9. > That is because the sequence 0.1, 0.11, 0.111, ... converges to 1/9 (with > precise definitions about what convergence does mean). > > In a similar way, ...111 represents a number in the n-adics. The > reason is that the sequence 1, 11, 111, ... converges. And so that > number is 1/(1-n) in the n-adics. Again, with precise definitions about > what convergence does mean. All that aside, each such string has a definite successor and predecessor, and can be ordered, so it acts as a number system. |