Notion Does Not Exhaust "unambiguously Described" Numbers
Not every number that we would informally say has been unambiguously described, is definable in the above sense. For example, if we can enumerate all such definable numbers by the Gödel numbers of their defining formulas then we can use Cantor's diagonal argument to find a particular real that is not first-order definable in the same language. The argument can be made as follows:
Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G(n) is the real number described by the nth description in the sequence. Using the diagonal argument, it is possible to define a real number x, which is not equal to G(n) for any n. This means that there is a language L' that defines x, which is undefinable in L.
Read more about this topic: Definable Real Number
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