An axiom is a statement that is taken as true without proof. They are the building blocks for all mathematics. Axioms are used to construct proofs for simple theorems, then those theorems build proofs for other, more complex, theorems.
The first rigorous construction of mathematics using axioms was in 300 BC, when Euclid, a Greek mathematician, published The Elements. Euclid used a set of five axioms to construct geometry - today called Euclidean geoemtry. David Hilbert expanded upon his work creating a set of 21 axioms that all geometry follows from. The next challenge was to take what had been done with geometry and extend it to all mathematics. Hilbert wanted to find out if there was a complete set of axioms on which all mathematics could rest.
Unfortunately, Hilbert's dreams were shattered when Kurt Godel published his two Incompleteness Theorems. The first proved that for any set of axioms there will always be something that cannot be proved with them. No matter what axioms you choose, you will always come to some problem, some theorem, that cannot be solved with them. The second stated that any consistent set of axioms cannot prove itself complete without an extra axiom being added.
This is where the Continuum Hypothesis lies. Since the beginning of the 20th century, mathematics rests on what is called the ZFC (Zermelo-Fraenkel set theory with Choice) - a collection of nine axioms for dealing with infinities. In 1940, Godel proved that a true Continuum Hypothesis is consistent with the ZFC axioms. Then, in 1963, Paul Cohen, an American mathematician, showed that a false Continuum Hypothesis is also consistent. That means that the Continuum Hypothesis is independent of the ZFC axioms, which means it has been proven unprovable.
Interestingly, this type of problem has happened before- when Euclid was trying to determine if parallel lines exist or not. A parallel line postulate was debated for centuries - it was unsolvable just like the Continuum Hypothesis is today. Mathematicians had a choice: assume parallel lines exist and get one type of geometry (Euclidean Geometry) or assume they don't and get a different geometry (hyperbolic or elliptical geometries). (See also: Euclid's Parallel Postulate) Both are valid options, although they have different results. The difference here is that axioms are supposed to be self-evident. Euclid's choice was straightforward - he could use his intuition. Infinities are a bit more complicated and so we are left with a more complicated choice: assume the continuum hypothesis is true and that the reals are \[\aleph_1\] or assume they don't. Both are valid, although one may be better than the other depending on the circumstances.
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