
Re: Mueckmeathical Misdefinitions
Posted:
Apr 21, 2017 9:05 PM


In trying to make sense of WM's posts, I consider that he is not using the set of real numbers most responders are using, given the cogent rebuttals.
For example, consider the value of the sum of .9 + .09 +.009... which is equal to .999..., which most responders would say absolutely is equal to the limit of 1, but WM may say that no matter how long you go, you cannot complete summing the infinite terms, so you will always have a value less than 1 (so there is a limit, but the expression is NOT equal to the limit; it is strictly less.)
Pedagogically, many people would look askew at such a statement, since categorically .999... = 1 in the real numbers, using most any common theory in use.
However, I beg the readers to consider an alternativethat WM is actually using the class of Surreal Numbers, where there is a limitting value of 1, but the value of .999... is strictly less than 1.
https://www.quora.com/Withtheexistenceofsurrealnumbershowcan0999999%E2%80%A61whentheremustbenumbersbiggerthan0999999%E2%80%A6butsmallerthan1
A more canonical source would be John H. Conway's "On Numbers and Games", second edition, page 43, in Chapter 4, On Algebra and Analysis of Numbers, subsection "Further remarks about analysis in No", where he says limits of partial sums... don't seem to work, where he gives a couple examples such as the limit of the squence 1,2,3... of all final ordinals cannot tend to omega, as there is a whole Host of numbers greater than any finite integer. For this reason, definitions of infinite sums has to be, in a certain sense, "global".
So don't pick on WM where he and Conway are in agreement, unless you think Conway is somehow wrong.

