> > > So what about sqrt(2)? or Pi? > > Are those real numbers? > > It depends on what we mean by real number. And yes, > I'm painfully > aware that many consider there to be one exact > specific definition and > nearly disallow all introspection into the idea > itself due to this.... > I regard them as real numbers because I have been > taught to. > If by "are they real" you mean can every other real > number be > expressed as strictly less or greater, than I would > say quite likely > and I currently hold it to be so. Yet I find this to > be seemingly true > about 0.999... as well and consider it in need of a > different label > than real number. > > The issue gets conceptually very complex for me. In a > sense, it seems > no real number can intervene between 0.999... and one > since decimal > cannot express one. But decimal cannot express a > whole range of real > numbers that exist between 0 and anything it could > ever hope to > express. > > If we accept ideas we have been taught, sqrt(2) and > pi will never be > able to explicitly located with a decimal > representation. I do not > believe an infinity of positions would explicitly > locate them either. > If you asked infinity itself to string digits > together in decimal > notation until it had an exact location of an > irrational number, I > doubt it would be able to comply in infinite > completion. Much as we > just can't divide 100 stones into 3 piles no matter > how we go about > it. This is something I have come to believe true. > For the same reason > 0.999... does not equal one, an infinite decimal for > any irrational > number would be strictly less as well - even if > astronomically beyond > our needs for accuracy. At no position within pi have > we explicitly > located pi, and should we have an infinity of digits > that accurately > represent the next needed increment in our system, I > am unable to > attribue an ability to the structure of infinity to > allow the final > infinite set to be any more explicit in its location. > > > > So why do you accept those infinite > > decimals as real but not .999... ? > > I do not accept infinite numbers to be real in my > current > understanding. They seem more a relation. For any > real number you > point out they will be more or less, but never > specifying an exact > location themselves. This is an interesting idea to > point if you just > step back and look at what it means to have an exact > location on the > number line, as between any two real numbers lie an > infinity of real > numbers infinitely nested. > > In my current personal definition of real number, a > real number > requires you stop at some point and say it's exactly > in one knowable > position. Many may say it is itself a real number, > but I do not feel > we can know other non-real relations such as 0.999... > to have > properties we would expect of real numbers. > > You'll have to excuse the implications of the ideas I > am sharing, as > they are new understandings that I feel in no way > relate to my proof > or are required by my proof. I share them only out of > a sense of > openess and exploration with you. > > > > > For that matter, what about the fraction 1/3? > > I'm sure you accept 1/3 as a real number. > > Do you accept 1/3 = .333... ? > > I believe 1/3 is exactly a real number. I believe any > division of a > span of the number line will yield a real number. > > I do not accept 1/3 = 0.333.... I regard 0.333... as > strictly less. In > my personal understanding, I could never hope to say > 0.999... does not > equal one while simultaneously saying 1/3 = 0.333... > so I am surprised > you would expect I might accept that as literally > valid. > > > Recall that the sum of an > > infinite series means, by definition, the limit of > the partial sums, > > providing the limit exists. > > The sum, a limit, is a definition of convention as it > yields a value > usable by us. To me a limit expresses a bound on such > an entity. It > does not dicate the nature of the entity itself. I > believe the limit > as the number of places grows in 0.999... is 1 in > terms of decimal > notation, but I believe an infinity of real numbers > must exist between > 0.999... and 1. > > I feel it important people undertstand the first > paragraph in my proof/ > analysis. Decimal notation has a huge range of real > numbers it cannot > ever hope to express. > > --charlie > > > > --charlie >
Yes, we know (by the Continuum Hypothesis, Cantor) that betweeness is infinite--not just "huge." Suppose we reject the CH? Every sequence is then finite and differs from another sequence that infinitely approaches, but does not reach it, by an infinitesimal margin. (This is the insight, of course, that led to the development of analysis--the study of continuous functions.)
Suppose we reject continuous functions? Because we we know that all real functions are continuous (Dedekind, Weyl, Brouwer, et al) then if you want to speak of a nonreal number less than unity and not differentiable from unity, it must exist on other than the real number line. Where then? In mathematics, as in physics, things that are not differentiable are identical.
Do you see yet what a tangled web you weave? If you insist that 1.0000... differs from 0.9999... where?
Certainly, you can invent your own mathematics. How useful is it? In some mathematical worlds, and certainly in yours, maybe such undefined concepts as "limit" and "function" don't "really" exist. What mathematics can we do without these concepts?
Even in ordinary arithmetic, there exist concepts undefined by the (Peano-Dedekind)axioms: "zero," "successor," and even "number" itself.
In the end, those of us who do standard mathematics accept the definition, 1.0000...= 0.9999..., not because we believe it to be true -- mathematics does not require one's personal belief -- but because it facilitates the construction of true statements consistent with the concepts of limit and function. That system is rich and productive. Is yours?