Every mathematical theory is founded upon a set of axioms. Axioms are the basic propositions of a theory and are used as starting points. They are simply assumed to be true; all other true statements must be derived from them. By starting with these basic assumptions it is possible to construct a coherent, consistent, and unambiguous theory.

It helps to compare a mathematical theory to a house. The axioms are standardized building blocks; rules of inference connect the blocks to each other - stacking, gluing, nailing, etc. Every part of the house - the windows, the door, the roof - is constructed out of the same basic blocks, and held together by the same methods of connection.

Mathematical arguments usually have a simple structure. Begin with a set of premises, then use established rules of inference to manipulate and combine these premises until you reach the desired conclusion...