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.