# Difference between axioms, theorems, postulates, corollaries, and hypothesis

I've heard all these terms thrown about in proofs and in geometry, but what are the differences and relationships between them? Examples would be awesome! :)

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Go read this Wikipedia article and the articles it links to. – kahen Oct 24 '10 at 20:22
One difficulty is that, for historical reasons, various results have a specific term attached (Parallel postulate, Zorn's lemma, Riemann hypothesis, Collatz conjecture, Axiom of determinacy). These do not always agree with the the usual usage of the words. Also, some theorems have unique names, for example Hilbert's Nullstellensatz. Since the German word there incorporates "satz", which means "theorem", it is not typical to call this the "Nullstellensatz theorem". These things make it harder to pick up the general usage. – Carl Mummert Oct 24 '10 at 23:15

A "hypothesis" is an assumption made. For example, "If $x$ is an even integer, then $x^2$ is an even integer" I am not asserting that $x^2$ is even or odd; I am asserting that if something happens (namely, if $x$ happens to be an even integer) then something else will also happen. Here, "$x$ is an even integer" is the hypothesis being made to prove it.