4- Compare/contrast the way we prove theorems in math (formalized informal reaso

Question

4- Compare/contrast the way we prove theorems in math (formalized informal reasoning) versus proving theorems in a formal deductive system such as Hilbert’s.
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?

Explanation / Answer

a precise and unambiguous description of the meaning of a mathematical term. It characterizes the meaning of a word by giving all the properties and only those properties that must be true.

Theorem — a mathematical statement that is proved using rigorous mathematical reasoning. In a mathematical paper, the term theorem is often reserved for the most important results.

Lemma — a minor result whose sole purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem. Very occasionally lemmas can take on a life of their own

a result in which the (usually short) proof relies heavily on a given theorem (we often say that “this is a corollary of Theorem A”).

Proposition — a proved and often interesting result, but generally less important than a theorem.

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