Consider the finite algorithm A, and the real number $0<T<1$. The output of A on input T is all possible theorems and provable propositions in ZFC, and only that.
Q1. Can such an algorithm and number exist?
Q2. Can we construct such an A and prove there exist such a T?
Q3. Can we prove or disprove that T is rational, irrational, normal, trancendental, algebraic?
Q4. Can T have a closed form expression?
Q5. Is it possible to prove or disprove that no finite algorithm that takes as input A, can output T to arbitrary precision?