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Melvyn Bragg and guests discuss an iconic piece of 20th century maths - G\\xf6del\\u2019s Incompleteness Theorems. In 1900, in\\xa0Paris, the International Congress of Mathematicians gathered in a mood of hope and fear. The edifice of maths was grand and ornate but its foundations, called axioms, had been shaken. They were deemed to be inconsistent and possibly paradoxical. At the conference, a young man called David Hilbert set out a plan to rebuild the foundations of maths \\u2013 to make them consistent, all encompassing and without any hint of a paradox. Hilbert was one of the greatest mathematicians that ever lived, but his plan failed spectacularly because of Kurt G\\xf6del. G\\xf6del proved that there were some problems in maths that were impossible to solve, that the bright clear plain of mathematics was in fact a labyrinth filled with potential paradox. In doing so G\\xf6del changed the way we understand what mathematics is and the implications of his work in physics and philosophy take us to the very edge of what we can know.With Marcus du Sautoy, Professor of Mathematics at\\xa0Wadham College, University of Oxford; John Barrow, Professor of\\xa0Mathematical Sciences\\xa0at the University of Cambridge and Gresham Professor of Geometry and Philip Welch, Professor of Mathematical Logic at the University of Bristol.