An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Dik T. Winter on 9 Sep 2006 22:30 You are running in circles. You have to show that *using my definition* it should be in the list. > > If it does not hold you should be able to give an index position > > such that it is false. > > It is not in the list, although we cannot give an index position which > is responsible for that fact. All possible indexes are in the list. Yes, it is not in the list. So what is the problem? All possible indexes are in the list, so: (1) A{p = digit position} E{q = list item} {such that q indexes p} which you deny. > > > > (2) A{p = digit position} E{q = list item} {such that q covers p} > > > > > > If your definition could be satisfied, the construction of the list > > > would imply this, yes. What we can safely say, however, is only: > > > (2') A{p = digit position of list item} E{q = list item} {such that > > > q covers p} > > > > The same here. But apparently you think my definition of 0.111... can not > > be satisfied. Why not? > > Because it is not in the list which, by definition, contains all > numbers which can be indexed and which can index. That is *not* the deinfition of your list. Your list contains *by definition* all finite sequences of 1. There is nothing in that definition that it contains all numbers that can be indexed and that can index. By extension, only the latter. > > > Now we may ask: Is it possible that a list item indexes or covers other > > > digits than those which are indexed or covered by list items? The > > > answer is: no. > > > > You are repeating yourself again, and I have already agreed to that. Why > > then repeat again and again? > > Because you assert that all digits of 0.111... can be indexed, which is > wrong. Slightly above you wrote: > All possible indexes are in the list. So that is also wrong? I have no idea what you mean with the distinction. > Indexing and covering "by all list numbers" is equivalent. Both > is true or both is false. Your statement is false. And I have no idea what the basis is for that statement. > > > > > And we may further ask: How can 0.111... be distinguished from all list > > > items? The answer is the only one possible: By digit positions occupied > > > by 1's which are neither indexed nor covered by list items. Why? > > > > This is false. 0.111... is distinguished from all list items in that it > > does not terminate. > > "Not to terminate" is not a property which can serve to distinguish a > number from others, because we can never observe the end (because it > does not exist) neither the non-end (because it is nothing but a > negation of an unobservable property). Again. We are arguing within the realm of the axiom of infinity. You are ignoring that axiom with that statement. With that axiom the set of natural numbers does exist, and so your list (which is also non-terminating) does exist, and so there is a distinction between the finite subsets of N and the set N itself. Although possibly not observable (whatever that may mean). > What can serve to distinguish > two numbers in unary representation is a 1 at a digit position. All > numbers which can be distinguished by such an 1 are in the infinite > list. Or whether a number ends in its unary representation or not. > Numbers are not processes. No. Please give a definition of the term "number". And a better one than you gave before (which included omega as a number). > > Consider them as decimal fractions. The list consists > > of the numbers (1-10^(-n))/9, 0.111... is 1/9. > > There is no n in the list (and in N) which yields 1/9. Therefore 1/9 = > 0.111... cannot completely be indexed. In that case there is an n in N such that the n-th digit of 1/9 does not exist. Or else, there is an n in N such that the n-th digit of 1/9 does exist, but that n does not exist. What do you mean? > > > because 0.111... does not have any other component. > > > > It does not have any other component, but also it does not terminate, > > in contrast to all numbers on the list. > > This cannot occur other than by more digits 1 than available by list > numbers. Why? Your list is non-terminating, so it can index non-terminating numbers. On the other hand, each list element is terminating. > > > So we may finally ask: Is it possible that all digit positions of > > > 0.111... are indexed or covered by list items? > > > The answer is obvious: no. > > > > Every digit position can be indexed and covered. Otherwise state which > > digit position can not be indexed, use my definition: > > Talking about 0.111... defined as: for every natural p digit p is 1, > > there are no other digtits. > > but there is no natural that covers all digits. > > > > And if you think that that definition is wrong, please *prove* that. > > It has been proven by showing that 0.111... does not belong to the set > of numbes which can be indexed completely. Wrong. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 9 Sep 2006 22:47 In article <1157836470.071804.254270(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <virgil-411702.16371406092006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: .... > A set of n elements is *a number*. It need no necessarily be a latin or > arabic symbol. Bizarre. What number is the set of all natural numbers? > > I count with my fingers. Does that count? > > The positions of your fingers are numbers. And you know what to do in > order to get from 32 to 33. I know what is the successor of the finger positions for 32. The position the 32 is all fingers (including the thumb) of the right hand up and two fingers raised on the left hand. So I just raise the ring-finger of the left hand to get the next number. No addition involved at all. And in both cases, the set contains either seven or eight raised fingers. > > And I can go to 35 with my > > fingers, and I use my feet (not toes) to extend to 143. I never really > > managed to do it base 2, but 4 and 128 are interesting numbers in that > > case. > > Every position of your fingers and feet is an expression of a number. Perhaps. > Counting without numbers would be nonsense. A boy like Virgil may count > 1, 2, 3, 3, 3, ... a, b, c, d, e, f, ... -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Tony Orlow on 10 Sep 2006 00:25 Dik T. Winter wrote: > In article <1157836470.071804.254270(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <virgil-411702.16371406092006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: > ... > > A set of n elements is *a number*. It need no necessarily be a latin or > > arabic symbol. > > Bizarre. What number is the set of all natural numbers? Hi Dik - Yes, it's bizarre to give that set a specific size, when it's unbounded. It's te "largest finite mantra". It's really not helpful. I won't go into the chanting part, because you won't get it, but Virgil knows what I'm referring to. > > > > I count with my fingers. Does that count? > > > > The positions of your fingers are numbers. And you know what to do in > > order to get from 32 to 33. To get to 32, we use the second hand. Has noone but me gotten Dik's joke about 4 and 128????? If Virgil had any sense of humor, he'd be peeved. Yeah, like he's not as a general rule....but whatever.... > > I know what is the successor of the finger positions for 32. The position > the 32 is all fingers (including the thumb) of the right hand up and two > fingers raised on the left hand. So I just raise the ring-finger of the > left hand to get the next number. No addition involved at all. > And in both cases, the set contains either seven or eight raised fingers. You start with the "ring" finger???/ No wonder you didn't get Dik's joke.... > > > > And I can go to 35 with my > > > fingers, and I use my feet (not toes) to extend to 143. I never really > > > managed to do it base 2, but 4 and 128 are interesting numbers in that > > > case. > > > > Every position of your fingers and feet is an expression of a number. > > Perhaps. As is every pile of sand, desert, and world. > > > Counting without numbers would be nonsense. A boy like Virgil may count > > 1, 2, 3, 3, 3, ... > > a, b, c, d, e, f, ... g - ALT - 103 one the numeric keypad. That leaves 152 more. Try ALT-255. :) It's the secret space. :D
From: Tony Orlow on 10 Sep 2006 00:30 David R Tribble wrote: > David R Tribble schrieb: >>> I really don't see where the physical limitation is for visualizing the >>> elements or the set. More to the point, I don't see how the phyics >>> of the real world have any limiting effect on abstract concepts. > > mueckenh wrote: >> What you are arguing is only the beginning of infinity, because you are >> unable to see more than this little realm. Try to determine the natural >> number which consists of the 10^100 digits following the first 10^1000 >> digits of pi. Your unability is not due to lack of time. This is only >> one of the infinitely many numbers which you cannot deal with (because >> it is not a number). > > Why is it not a number? It has all the mathematical properties > required of a number. Having "all its digits enumerated" is not a > required property. Hi David - :) What IS a number, and what IS mathematics? When you can answer one or both of these questions, please do. > > Supposing that I can in fact determine all 10^100 digits of this > number. What then? Does that number now magically possess > an existence that countless others do not, i.e., has it suddenly > become a "more real" number than, say, the next 10^100 digits > of pi? I agree with this point David. There is no finite limit to numbers. But, I'd still like answers to the preceding questions. > > Likewise, until I actually enumerate the digits of some specific > number, are you saying it does not (yet) exist? If that's the > case, does pi or e exist? > It's the pure ratio. Don't we all know that? Beautilicious, Tony
From: Virgil on 10 Sep 2006 00:37
In article <45039397(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Dik T. Winter wrote: > > In article <1157836470.071804.254270(a)i3g2000cwc.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > A set of n elements is *a number*. It need no necessarily be a latin or > > > arabic symbol. > > > > Bizarre. What number is the set of all natural numbers? > > Hi Dik - > > Yes, it's bizarre to give that set a specific size, when it's > unbounded. That depends entirely on how one defines "size" of a set. If one defines it in terms of bijectability with other sets, as is the reasonable definition for finite sets, there is no problem. > It's te "largest finite mantra". Only in TO's form of self worship. |