Application of the Gödel’s system

Using nothing but Gödel’s system P, prove that LaTeX: a\:\vee\neg aa ∨ ¬ a is true for all LaTeX:
(5 Points) Construct a complete formal system of proof and prove that it is complete. (This one is actually quite
easy, so when you discover the answer do not tell anyone!)
Prove or disprove the following propositions.
(10 Points) The set of all finite but unbounded sequences of the form LaTeX: \left\langle
x_1,x_2,\ldots,x_n\right\rangle⟨ x 1 , x 2 , … , x n ⟩ where LaTeX: x_i\in\mathbb{N}x i ∈ N and LaTeX:
n\in\mathbb{N}n ∈ N is countably infinite.
(15 Points) The set of all formal proof systems is countably infinite.
(10 Points) Given any consistent proof system, adding an undecidable proposition as an axiom still yields an
incomplete system of proof no matter how many iterations you perform. That is to say, consistent proof systems are recursively incomplete.

Sample Solution