On the computability of a family of "universal" real numbers - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T13:37:55Z http://mathoverflow.net/feeds/question/3411 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/3411/on-the-computability-of-a-family-of-universal-real-numbers On the computability of a family of "universal" real numbers Halfdan 2009-10-30T06:21:24Z 2009-11-01T04:14:44Z <p>Let g(i) be the minimal Goedel sequence (generated by a suitable decimal encoding scheme x) in bijective correspondence with the computable real number n(i), so that the sequence g(1), g(2),... is in ascending order by length of its member sequences.</p> <p>Informally, g(i) is the minimal (finite) length "Program" sequence representing the recursive function, which defines the value of n(i) with i being its corresponding Goedel number.</p> <p>We define the number Ux as the number with the decimal expansion 0.g(1)g(2)..... We have thereby constructed a number in the domain [0;1] which encodes the set of all computable real numbers with an associated total order.</p> <p>The real number Ux is clearly definable, non-random, and most certainly transcendental. But is it computable?</p> <p>Throwing the question out there for inspiration...</p>