Gödel's incompleteness theorem for formal arithmetic

In the history of development of mathematics a very important achievement is Gödel's incompleteness theorem. This result radically changed the main direction of mathematical researches. The beginning of 20th century may be regarded as a crisis period in mathematics, because a lot of new paradoxes and open problems have appeared. But in 1931 Gödel published his incompleteness theorems in "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme" (in English "On Formally Undecidable Propositions of Principia Mathematica and Related Systems"). In that article he proved that: (a) if a mathematical theory extends formal arithmetic and is recursively endurable, than it is incomplete; (b) it is impossible to show the consistency of arithmetic by means of only finitary methods.