On Friday, January 11, 2013 4:16:39 AM UTC-5, zuhair wrote: > Lets say that a real r is finitely definable iff there is a predicate P that is describable by a Finitary formula that is uniquely satisfied by r. Formally speaking: r is finitely definable I think this would be more helpful if "finitely definable" were defined more carefully. The first stab would be to say that it means computable. ie there is a Turing machine that will compute the real. This is not adequate; there are recursively enumerable sequences of computable numbers taht do not have a computable LUB. So; extend the definition. Assume that one has access to an oracle that can solve the halting for Turing Machines. Unfortunately this only moves the problem. One finds one needs a more poerful oracle to solve the new halting problem - and so on. This hierarchy is called the Kleene hierarchy. The set of real numbers thus defined is called the set of hyper arithmatic numbers. (For a fuller, better description , search the web. Wikepedia has a definition.) I believe that the set of hyperarithmaqtic reals is what you mean by finitely definable. Dick
