ABSTRACT
This chapter examines the idea that the axiomatic method embodies the way we express our arguments and conversations. The notions of explaining or understanding entail the ideas of translating, of interpreting. On the basis of intuition then, and the consistent application of the rules learned from it, mathematical knowledge can be expanded far beyond what is merely intuitive. Mathematics, has a strong axiomatic structure. Kurt Godel theorem is derived on the basis of a diagonal argument. The characteristic feature of this type of reasoning is the formulation of propositions in a self-referent manner especially conceived to upset a certain assumption. Primitive recursive functions are those that involve predictable terminating calculations. A formal system is made by symbols and rules establishing the basis for the legal production of further strings of symbols. Godel showed that any search for a string having a special typographical property has an arithmetical equivalence to the search for an integer with a corresponding special arithmetical property.
