There are, however, some statements that are entirely un-decidable; even with extra information - or indeed, with every piece of information possible - these statements cannot be determined to be true or false. The most famous example is: 'This statement is false.' If it's true, it's false; if it's false, it's true. It is a perfectly well-formed, grammatically correct statement for which we cannot assign a true or false value.

These are the sorts of statements to confuse children but they proved to be an anathema to mathematicians, whose purpose in life is to determine whether or not well-formed mathematical expressions are true or false. If statements such as 'this statement is false' have analogues in the language of mathematics, then there are expressions that cannot be decided - and so the mathematicians have set themselves an impossible task. This became important at the start of the 20th Century, when mathematicians such as David Hilbert and Kurt Gödel tried to establish mathematics on the most rigorous possible foundations; they were trying to prove that mathematics was complete and self-consistent.

Unfortunately, they managed instead to prove that it wasn't - but on the road to that discovery, they managed to invent computers, help to win World War II and built our modern 'information age'.

Read this full article at Silicon.com

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