EDIT: This question is NOT about 0.9999... !!! It is clearly stated in the heading. It is about definition of numbers. The confusion in all the comments and the answer - assuming it is about 0.9999... - is related to the confusion about that number. I admit the question is fairly broad. However, philosophical questions tend to be broad.
Let us consider again - perhaps legal in this rule-laden environment, perhaps not - the question: ” Is 0.9999… a number, and if it is, is it perhaps =1?
This is actually not my question but I am surprised I don’t se the answer that comes to my mind: 0.9999… is not a number but a series with the limit 1. But if you insist: Can we accept the notion of numbers such as those depending on an infinite series of decimals? Obviously we can get around some repetitive numbers by representing them with the limit of a series, but how about infinite numbers in general?
I am intrigued with the notion of the definition of real numbers, and I note that quite a few questions is related to this issue (even without my own follow up question "Constructivism implied or not").
1 All real numbers can be expressed as a limit if rational numbers? 2 0.9999…=1 in several versions – related in my mind but perhaps not explicitly. 3 Are real numbers “a joke”?
I have met two conflicting points of view
i) is my own after having consulted (as I believe) Gödel-specialist Prof Podnieks who informs me that
“Any Goedel-style enumeration can cover only those real numbers that are definable by formulas (in some fixed language). Thus, if in one's set theory, there are uncountably many real numbers, then some of these numbers must be undefinable (by formulas)” (private communication).
ii) Hamkins in http://mathoverflow.net/questions/44102/ referred to in question 3 above: “The concept of definable real number, although seemingly easy to reason with at first, is actually laden with subtle metamathematical dangers to which both your question and the Wikipedia article to which you link fall prey. In particular, the Wikipedia article contains a number of fundamental errors and false claims about this concept./ …. / The naive account continues by saying that since there are only countably many such descriptions φ, but uncountably many reals, there must be reals that we cannot describe or define. But this line of reasoning is flawed in a number of ways and ultimately incorrect. The basic problem is that the naive definition of definable number does not actually succeed as a definition.”
My problem is that Gödel’s enumeration looks very similar to me, and that has succeeded as a definition.
Many articles deals with theproblem, such as “How real are real numbers?” from http://www.cs.auckland.ac.nz/~chaitin/olympia.pdf but I am not sure it provides much of an answer.
In any case, I speculate that these - seemingly – conflicting views are the root of problem. It would be valuable to have an answer or something in that direction. Better understanding would perhaps not settle all these questions, but it would probably take some of the confusion out of the debate.