Wikipedia 10K Redux by Reagle from Starling archive. Bugs abound!!!

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Mathematical theorem proven by [[Kurt Goedel]. Somewhat simplified, this states that in any axiomatic system sufficiently complex to allow one to do mathematics one can construct a statement that can be neither proved nor disproved within that system. 
In effect, Godel's proof consisted of formally constructing the statement "This statement is false" within the system. 

The combined work of Godel and [[Paul Cohen]] has shown that in standard Set Theory the truth or falsity of the [[axiom of choice]] and [[continuum hypothesis]] are similarly undecidable.