Proof That Computers Can't Do Everything (The Halting Problem)
From IST Learning Design
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This video gives an informal presentation of Alan Turing's Halting Theorem, a serious, highly influential result in computer science.
A few more comments on this video:
1) This video skips a lot of technicalities for sake of simplicity. There are many rigorous descriptions of this proof easily found on the web.
2) There really is an unbeatable checkers machine. See here: http://en.wikipedia.org/wiki/Draughts#Computer_draughts