Proving Godel's Theorem, a set-theoretical approach

Before Godel's incompleteness theorems, logicians such as Bertrand Russel and Alfred Whitehead pursued an ideal axiomatic system which would have created a reliable framework to successfully prove or refute every mathematical sentence. Godel proved that such systems can never be created. In fact, Godel's incompleteness theorems establish that axiomatic systems that are complex enough to formulate arithmetic can never generate a proof of all the logical statements that are expressible inside of them. According to the first incompleteness theorem, there are constructible mathematical sentences that can never be proven to be true or false using the axioms and the logical rules of the system. Furthermore, the second incompleteness theorem argues that the consistency of all axiomatic systems which contain Peano or Robinson arithmetic can never be determined using the rules and the proof mechanisms available in the system.