Definable Real Number - Notion Does Not Exhaust "unambiguously Described" Numbers

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

Famous quotes containing the words notion, exhaust and/or numbers:

    Some ne’er advance a judgment of their own,
    But catch the spreading notion of the town;
    Alexander Pope (1688–1744)

    Only the really plain people know about love—the very fascinating ones try so hard to create an impression that they very soon exhaust their talents.
    Katharine Hepburn (b. 1909)

    All experience teaches that, whenever there is a great national establishment, employing large numbers of officials, the public must be reconciled to support many incompetent men; for such is the favoritism and nepotism always prevailing in the purlieus of these establishments, that some incompetent persons are always admitted, to the exclusion of many of the worthy.
    Herman Melville (1819–1891)