The Incompleteness Theorem , proved by Kurt Gödel and published in 1931, actually showed that there exist logical statements whose truth value is undecidable, that is, they cannot be proved either true or false. The following was taken from the website "What is Incompleteness Theorem? - Definition from WhatIs.com".
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The Incompleteness Theorem is a pair of logical proofs that revolutionized mathematics. The first result was published by Kurt Gödel (1906-1978) in 1931 when he was 24 years old.
The First Incompleteness Theorem states that any contradiction-free rendition of number theory (a branch of mathematics dealing with the nature and behavior of numbers and number systems) contains propositions that cannot be proven either true or false on the basis of its own postulates.
The Second Incompleteness Theorem states that if a theory of numbers is contradiction-free, then this fact cannot be proven with common reasoning methods.
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