On Mar 3, 1:05 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > On Mar 2, 3:01 pm, Graham Cooper <grahamcoop...@gmail.com> wrote: > > > > > > > > > > > On Mar 3, 7:35 am, "Ross A. Finlayson" <ross.finlay...@gmail.com> > > wrote: > > > > On Mar 2, 12:30 pm, Graham Cooper <grahamcoop...@gmail.com> wrote: > > > > > On Mar 2, 7:28 pm, Rupert <rupertmccal...@yahoo.com> wrote: > > > > > > On Friday, March 1, 2013 9:39:31 PM UTC+1, Graham Cooper wrote: > > > > > > A LIST oo ROWS LONG! > > > > > > > 0.00... > > > > > > > 0.00... > > > > > > This post is incoherent dribble. > > > > > I showed a partial infinite list of reals. > > > > > 0.00.. > > > > 0;00.. > > > > .. > > > > > Do you have an ANTI-DIAGONAL function to support your claims it is > > > > incomplete? > > > > > What is your anti-diagonal function? > > > > > Herc > > > > How would you establish that the expansions you begin to detail would > > > map on to any segment of R? > > > > Well, that gets into whether the function, that makes a list these > > > expansions, has as a range, an interval of reals. > > > > So, look at the equivalency function, as I call it, it's quite well- > > > defined, it goes to one, and in binary there's only one anti-diagonal, > > > and it's one. > > > > I'll agree that a more carefully defined function, that would have as > > > each initial segment of each initial segment, of a matrix of values of > > > the expansions, zeroes, with the only anti-diagonal in binary being > > > ones, with real value one, may go from zero, to one. > > > > And: only one does. > > > > Then, for a conscientious mathematician, formalist year-round, that's > > > compelling. > > > > There are lots who would work in foundations, but transfinite > > > cardinals aren't used in real analysis, or continuum analysis for > > > applications and physics. And, physics needs new methods to explain > > > results of experiment. And, results in the digital are available via > > > asymptotics. Good day. > > > > Regards, > > > > Ross Finlayson > > > So your argument is: > > > ---------------------------------------- > > > Given this sample of a list oo rows long IN BINARY > > > 0.00.. > > 0.00.. > > .. > > > We know 0.11.. > > is missing from the List by Extrapolation over all digit positions. > > > ---------------------------------------- > > > Given this sample of a list oo rows long IN TERNARY > > > 0.00.. > > 0.00.. > > .. > > > We know > > 0.11.. > > 0.12.. > > 0.21.. > > 0.22.. > > > are missing from the List by Extrapolation. > > > ---------------------------------------- > > > Given this sample of a list oo rows long IN BASE 4 > > > 0.00.. > > 0.00.. > > .. > > > We know > > 0.11.. > > 0.12.. > > 0.13.. > > 0.21.. > > 0.22.. > > 0.23.. > > 0.31.. > > 0.32.. > > 0.33... > > > are missing from the List by Extrapolation. > > > ---------------------------------------- > > > Since all above arguments must hold, the latter more absurd ones are > > enough to throw doubt on Cantors Method - which is actually just > > induction over ALL sizes of FINITE lists, since no_new_digit_string is > > calculated in the Anti-Diagonal on some infinite lists of reals. > > > Herc > > --www.BLoCKPROLOG.com > > Formal rigor is a goal of foundations. > > Here then a consideration of the information content in various bases > or radices used for expansions (of fixed magnitude precision) has that > while any finite base can be used to represent any given (standard) > real number between zero and one in an infinite sequences of elements > of that base, it takes more information to store elements in a higher > base. In this sense binary gets the most elements of the expansion > for a given range of precision. For example with twenty bits in > binary it takes only 2^20 distinguished elements to comprise it, for > twenty bits in decimal, 10^20 which is already 5^20 times as great as > 2^20. Thus as there is approximated a real value to a given number of > digits, that precision is reached first in binary. Then in a sense, > writing the digits in binary is the most efficient way to represent > the number. > > Then for this equivalency function, representing the numbers in > ternary, quaternary, and so on, bases > 2, has that the antidiagonal > could be at minimum .111_b, or sum_i=1^oo b^-i, the sum of negative > powers of b, or 1/(b-1). Then as the base goes from 2 onward, the > interval would range to at most 1/2, 1/3, .... Yet, with an infinite > radix, it would go back to being one. > > This loophole, this special case, this prototype of functions defining > intervals as arrayed lines of points in drawing them as arrayed from > point to point, has simply that Cantor's results on reals are that > they are only arrayed this way, countably. These are for the number- > theoretic considerations, for the set-theoretic considerations there > are as well notions of what the real numbers are as extra "complete" > ordered field, or what the linear continuum is, besides as of the > "well-founded" infinity. > > So, for a function that has each initial segment of an expansion, for > each initial segment of expansions, having at each modulus zero, and > that the only symbol in the language for all bases not zero is one, > the range of this particular function goes to one, which is greater > than or equal to the value everywhere different of each of the initial > segment of each of the initial segment, and simply scaling the range > (of EF) to the reciprocal of the base sees the anti-diagonal: nowhere > before the end of the list, which it is. The limit IS the sum. > > For formalists, "exists" is particular. > > Regards, > > Ross Finlayson
You are still left with the Absurdity that given only the following information
0.00... 0.00.. ..
in Decimal, 80 different reals are calculated USING_CANTORS_METHOD that are "missing" from that List.
The Method applied is "anything-but-that-digit" or NOT(DIGIT) which does look more rigorous in Binary, but you don't get to select which base a general proof on reals looks ok in.
I am expanding logic into Natural Language and the central reasoning heuristic seems to be
NOT ALL == SOME DON'T
!(ALL(X) p(X)) <-> EXIST(X) ~p(X)
I already programmed Prolog to solve that Not All Natural Numbers are Even by finding a Natural Odd number s(0), a counterexample.
e(A, not(evens)) <- e(A, odds)
I'm now looking at : NOT(EVERYTIME(X)) event(...) <-> SOMETIMES( Opposite(event)) (NOT(EASY) method) <-> SOMETIMES(hard, method)