n22 - CS 70 Discrete Mathematics and Probability Theory...

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Unformatted text preview: CS 70 Discrete Mathematics and Probability Theory Fall 2010 Tse/Wagner Note 22 Self-Reference and Computability The Liars Paradox Propositions are statements that are either true or false. We saw before that some statements are not well defined or too imprecise to be called propositions. But here is a statement that is problematic for more subtle reasons: All Cretans are liars. So said a Cretan in antiquity, thus giving rise to the so-called liars paradox which has amused and confounded people over the centuries. Actually the above statement isnt really a paradox; it simply yields a contradiction if we assume it is true, but if it is false then there is no problem. A stronger formulation of this paradox is the following statement: This statement is false. Is the statement true? If the statement is true, then what it asserts must be true; namely that it is false. But if it is false, then it must be true. So it really is a paradox. Around a century ago, this paradox found itself at the center of foundational questions about mathematics and computation. We will now study how this paradox relates to computation. Before doing so, let us consider another manifestation of the paradox, created by the great logician Bertrand Russell. In a village with just one barber, every man keeps himself clean-shaven. Some of the men shave themselves, while others go to the barber. The barber proclaims: I shave all and only those men who do not shave themselves. It seems reasonable then to ask the question: Does the barber shave himself? Thinking more carefully about thereasonable then to ask the question: Does the barber shave himself?...
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This note was uploaded on 12/08/2010 for the course CS 70 taught by Professor Papadimitrou during the Fall '08 term at Berkeley.

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n22 - CS 70 Discrete Mathematics and Probability Theory...

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