| In the usual setting in which we do mathematics, this is a well-defined real number. That it's hard to determine its digits (even impossible when restricted to use an algorithm) is of no relevance. However, you might be interested in the effective topos. This is an alternate mathematical universe in which one can't define construct this real number. It has a number of curious properties: * The statement "1 + 1 = 2" is true in that topos, just like it is in the ordinary topos. * The statement "any number is either prime or not prime" is true in that topos, just like it is in the ordinary topos, however its truth is not trivial. * The statement "any real number is either equal to zero or not equal to zero" is false in that topos. * The statement "any function is computable" is true in that topos (and wildly false in the standard topos). * The statement "any function is continuous" is true in that topos (and wildly false in the standard topos). Details are in this set of slides: https://rawgit.com/iblech/mathezirkel-kurs/master/superturin... Questions are welcome! |