On May 31, 5:12 am, chaja...@mail.com wrote: > > 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....
There are very good reasons that the standard definition of the real numbers is used. The real numbers are the completion of the rational numbers. They form a ordered field (i.e. you can add, subtract, multiply, divide by anything other than 0, and the order of elements acts exactly as you would expect). Note that you can define the real numbers without ever talking about decimals. Decimal notation is just one way of describing the real numbers.
> 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.
No, by "are they real" I mean much more. As well as an order you also need the field operations (addition, subtraction, multiplcation and division) and you need them to preserve order.
> 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. >
It is not difficult to put an ordering on decimal notation such that 0.999... comes exactly where you expect it (before 1, but after every other decimal notation starting 0.). But this ordering is not the ordering of an ordered field.
> 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.
No. If we use the standard definition of real numbers and the standard way of talking about decimals (infinite decimals are defined in terms of limits), then decimals can express any real number (including sqrt(2) and pi).
> > 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.
It was to avoid nonsense like "would be able to comply in infinite completion" that the concept of limits was developed.