> 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.