# Does this number exist?

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?

-
What exactly do you mean by giving a real number as input to a finite algorithm? Since the sentences of the language of set theory are enumerable, there is a real number between $0$ and $1$ that has a $1$ in each binary digit corresponding to a theorem of ZFC and a $0$ everywhere else -- if you "give this number as input" to an algorithm that examines its digits one by one, you get the desired output -- but what would that mean? – joriki Apr 7 '11 at 0:38
You can use Chaitin's constant to encode all this information, for instance. Note however that this number is necessarily noncomputable (so in particular, must be transcendental, and the answer to Q5 is no), as the halting problem (or first-order logic) is undecidable. – Akhil Mathew Apr 7 '11 at 0:43
So we know such a number exist, how can we know that for instance e/Pi does not by some magical coincidence have this property? – TROLLKILLER Apr 7 '11 at 0:59
I'm having trouble with 'exists' (I don't recognize it as a technical term here). Do you mean computable? What about $T$? DI'm guessing you intend that it 'exists'? Is it also computable? I feel like that should be part of the prerequisites to the question. – Mitch Apr 7 '11 at 2:50