Godel's Incompleteness Theorems

In Our Time: Philosophy

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4.5 • 1.3K Ratings

🗓️ 9 October 2008

⏱️ 42 minutes

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Summary

Melvyn Bragg and guests discuss an iconic piece of 20th century maths - Gödel’s Incompleteness Theorems. In 1900, in Paris, 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 – 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 Gödel. Gödel 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 Gödel 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 Wadham College, University of Oxford; John Barrow, Professor of Mathematical Sciences at the University of Cambridge and Gresham Professor of Geometry and Philip Welch, Professor of Mathematical Logic at the University of Bristol.

Transcript

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0:00.0	Thanks for downloading the In Our Time podcast.
0:02.2	For more details about In Our Time and for our terms of use, please go to BBC.co.
0:07.1	UK forward slash Radio 4.
0:09.4	I hope you enjoy the program.
0:10.7	Hello in 1900 in the German city of Kurnig'sburg, the the The edifice of maths was grand and ornate, but its foundations, called axioms,
0:24.4	were shaking with inconsistency and lurking paradox.
0:27.9	And so at that conference, a brilliant young German mathematician called David Hilbert
0:32.3	set out a plan to rebuild them, to make them consistent, all-encompassing,
0:36.7	and without any hint of a paradox.
0:38.8	Hilbert was one of the greatest mathematicians that ever lived, but his plan failed spectacularly and it did so because
0:44.8	of the incompleteness theorems these are the work of Kurt Gerdle and they changed the
0:49.9	way we understand maths took us to the very limits of logic and sent challenges
0:54.1	spilling out into the world of physics, philosophy and beyond. With me to discuss
0:58.4	girdles in completeness theorems, and John Barrow, professor of mathematical
1:02.1	sciences at the University of Cambridge and Gresham
1:04.2	Professor of Geometry, Philip Welsh Professor of Mathematical Logic at the University of Bristol,
1:09.0	and Marcus Jusotoy, Professor of Mathematics at Wadham College, University of Oxford.
1:14.0	Marcus Otoy, as I mentioned in introduction,
1:16.0	Foundations of mathematical systems are called axioms.
1:19.0	So perhaps you could give us foundations for this program
1:21.0	by explaining what axioms are.
1:23.0	Yeah, this goes really to the heart of what mathematics is about.
	...

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