1- What is a deductive system? What were they originally developed for? 2- What is the Hilbert deductive system? 3- What is the main difference between the Hilbert system and the Gentzen system? 4- Compare/contrast the way we prove theorems in math (formalized informal reasoning) versus proving theorems in a formal deductive system such as Hilberts. 5- Proofs in the Hilbert system could be elaborate; how can the process be simplified? 6- What is the importance of the deduction rule? 7- Explain with examples how the axioms of the Hilbert system are applied in proofs. 8- In the context of deductive systems, what do the following terms mean: consistency, independence, soundness, completeness?