# Can non-computable real numbers be defined without making use of any notions from computability theory

I am not sure if this type of question is appropriate for "mathoverflow.net", but I will take the chance. Are there any examples of (well-known or interesting) problems in geometry which ask for the minimum (or greatest lower bound) or else ask for the maximum (or least upper bound) of a set of real numbers such that the answer is known to be non-computable or (at least) has not been proved to be computable? I am thinking, for instance, of the following modified version of Lebesgue's Universal Covering problem. "What is the minimum (or greatest lower bound) of plane convex sets that cover every plane set having diameter 1" I want to know about definitions of specific non-computable real numbers which do not make use of any concepts from computability-theory, such as Turing machines or Recursive Functions. Of course, once you have the definition, you may use these concepts to prove that the number you have defined is non-computable. I feel that geometry is one branch of mathematics in which there might be a good chance of encountering such definitions.

• See mathoverflow.net/q/11540/1946 for numerous examples of non-computable problems, each of which can be viewed as a non-computable real number (if one thinks of the binary digits), and most of those do not explicitly involve concepts from computability. Indeed, several of the problems, such as the tiling problems or the problems concerning manifolds and homotopy equivalence, have what might reasonably be deemed a geometric aspect. – Joel David Hamkins Sep 21 '15 at 19:54
• Thanks for this reference. I will try to formulate some of these examples as geometric problems with specific real number solutions. – Garabed Gulbenkian Sep 22 '15 at 20:42
• My statement of the modified Lebesgue covering problem is not quite correct. It should read "What is the minimum (or greatest lower bound) of the set of diameters of plane convex sets that cover every plane set having diameter 1" – Garabed Gulbenkian Sep 22 '15 at 20:52