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Axioms are statements that form the foundation of a field of mathematical study. A mathematical proof is simply a logical argument that shows that a result follows from the chosen set of axioms. The axioms themselves cannot be proved. The set of axioms must not lead to contradictory conclusions, nor should it be possible to prove one axiom from others in the set.

Axioms do not have to be self-evidently true. They need only be self-consistent. However, many of the possible sets of axioms lead to uninteresting and barren fields of study. Axiom choice is not therefore completely arbitrary.

An interesting case is the set of axioms needed to define the study of geometry. Euclid used five postulates - along with some definitions and common notions - as the foundation for the geometry which we now know as Euclidean Geometry. For years it was conjectured that the fifth postulate, the parallel postulate, was superfluous. Many people attempted to prove it and failed. However, when alternatives to the fifth postulate were finally explored, new fields of study emerged such as spherical and hyperbolic geometries. Euclidean Geometry, though still very important, is now seen as just one of many possible geometries. Einstein used non-Euclidean geometries to develop his successful theory of General Relativity.

Here are the 4 axioms of Group Theory: Given a set G together with a binary operation °: G ×G → G,

• There exists a unique e ∈ G such that g °e = e °g = g for all g ∈ G;
  
• For all g,h ∈ G, then g °h ∈ G;
  
• For each g ∈ G, there exists an element, known as g⁻¹, where g °g⁻¹ = g⁻¹ °g = e;
  
• For all f,g,h ∈ G, f °(g °h) = (f °g) °h.