Can N be constructed?
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From: Virgil on 13 Jun 2010 19:35 In article <58445485-8c41-4a8f-b89a-208fc4d7755a(a)r27g2000yqb.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 13 Jun., 20:05, Virgil <Vir...(a)home.esc> wrote: > > In article > > <93a79992-2f0a-4e51-8256-203a1396f...(a)i28g2000yqa.googlegroups.com>, > > > > > > > > > > > > �WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 12 Jun., 22:05, Virgil <Vir...(a)home.esc> wrote: > > > > > > > Correct. But 1/9 can be constructed, i.e., every digit can be > > > > > determined by an algorithm. Same holds for sqrt(2) and pi. But the big > > > > > error was to assume that an infinite sequence could also exist without > > > > > an algorithm and define a number. The number would only be defined > > > > > when the last digit of the sequence was known. > > > > > > How does one ever "know" the "last digit" of a sequence which does not > > > > have a last digit, for example, 0.333...? > > > > > Never. > > > > > > Nevertheless, there is a well-defined number in decimal arithmetic > > > > defined by that sequence. > > > > > No, not by that sequence. The number is defined by a word, in there > > > are many words, each defining it: > > > 1/3 > > > 0.333... > > > one over three > > > ein Drittel > > > 2:6 > > > > > There is no infinite sequence defining it. > > > > There is in my world. It is a shame hat WM's world is so much smaller. > > Every term of a sequence "counts". Every term of a sequence must be > known, if the number defined by that sequence is to be known. (An > unknown definition is not a definition!). For every positive rational, there is an "infinite decimal" which can be entirely defined from the ratio of naturals, in simplest form, representing that rational. Thus EVERY DIGIT of its decimal expansion can be known, and from that known expansion the original ratio of naturals can be recovered. > But it is impossible to know > every term of an infinite sequence Not for base n expansions, n > 1, of positive rationals. > > What you think or believe or even have learned in school is wrong. > A definition D defines a sequence S: D ==> S. > In your world people think that would imply S ==> D. > > That is an error. Certain sequences define rationals just as exactly as anything else can define them. Though perhaps less conveniently for arithmetic. > > > > > > Only from the finite > > > definition, you can obtain every digit of the sequence. > > > > And from any such sequence, one can find a number > > Yes. But then you don't need the sequence. Who uses an infinite array > of 3's instead of 1/3? The point being that EITHER 1.3 or 0.333..., can logically be used as the definition of a certain rational number, though equally conveniently. > > You would not be able to obtain a number from the sequence > 0.111111111111111111111111111111111111111111111111111111111111 here I > stopp, but even from 10^100^10000 digits you could not say what number > is meant in the end. If one knew the base of the n-ary system being used and knew that the sequence either ended with a fixed number of digits, to did not ever end, in either case, a particular rational has been defined. > > Sorry, probably you will not be willing to grasp that, and will > continue to believe in uncountability and implication reversal. > Nevertheles, this is one of the most spectacular mistakes ever > committed in history of mankind. Allowing WM to try to teach mathematics is a far worse m
From: WM on 14 Jun 2010 08:50 On 14 Jun., 01:35, Virgil <Vir...(a)home.esc> wrote: > > Yes. But then you don't need the sequence. Who uses an infinite array > > of 3's instead of 1/3? > > The point being that EITHER 1.3 or 0.333..., can logically be used as > the definition of a certain rational number, though equally conveniently. Yes because both are finite definitions. > > > > > You would not be able to obtain a number from the sequence > > 0.111111111111111111111111111111111111111111111111111111111111 here I > > stopp, but even from 10^100^10000 digits you could not say what number > > is meant in the end. > > If one knew the base of the n-ary system being used and knew that the > sequence either ended with a fixed number of digits, to did not ever > end, in either case, a particular rational has been defined. If you knew that! How could you get to know it? By a finite definition. But then the infinite sequence was no longer needed. In fact. It is not needed at all. You are not able to comprehend that, and that will not change for ever. Therefore EOD. Regards, WM
From: Virgil on 14 Jun 2010 15:45 In article <634d7e26-04c3-495b-828c-74ca4bbcd533(a)i28g2000yqa.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 14 Jun., 01:35, Virgil <Vir...(a)home.esc> wrote: > > > > Yes. But then you don't need the sequence. Who uses an infinite array > > > of 3's instead of 1/3? > > > > The point being that EITHER 1.3 or 0.333..., can logically be used as > > the definition of a certain rational number, though equally conveniently. > > Yes because both are finite definitions. > > > > > > > > > You would not be able to obtain a number from the sequence > > > 0.111111111111111111111111111111111111111111111111111111111111 here I > > > stopp, but even from 10^100^10000 digits you could not say what number > > > is meant in the end. > > > > If one knew the base of the n-ary system being used and knew that the > > sequence either ended with a fixed number of digits, to did not ever > > end, in either case, a particular rational has been defined. > > If you knew that! How could you get to know it? By a finite > definition. But then the infinite sequence was no longer needed. In > fact. It is not needed at all. It is not a question of necessity but of sufficiency. Once you have it, it suffices.
From: George Greene on 14 Jun 2010 17:06 On Jun 14, 8:50 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > If you knew that! How could you get to know it? By a finite > definition. But then the infinite sequence was no longer needed. In > fact. It is not needed at all. Infinite strings ARE SO TOO needed for rationals with denominators that have factors that don't occur in the base. Yes, you could use a finite program, but it would be a finite program that generated INFINITE output.
From: David R Tribble on 14 Jun 2010 19:55
Virgil wrote: >> How does one ever "know" the "last digit" of a sequence which does not >> have a last digit, for example, 0.333...? >> Nevertheless, there is a well-defined number in decimal arithmetic >> defined by that sequence. > WM wrote: > No, not by that sequence. The number is defined by a word, in there > are many words, each defining it: > 1/3 > 0.333... > one over three > ein Drittel > 2:6 > > There is no infinite sequence defining it. Only from the finite > definition, you can obtain every digit of the sequence. You left out a few more: 3/10^1 + 3/10^2 + 3/10^3 + 3/10^4 + ... 1/4^1 + 1/4^2 + 1/4^3 + 1/4^4 + ... |