n22 - CS 70-2 Spring 2009 Discrete Mathematics and...

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CS 70-2 Discrete Mathematics and Probability Theory Spring 2009 Alistair Sinclair, David Tse Note 22 Self-Reference and Computability The Liar’s 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 liar’s paradox which has amused and confounded people over the centuries. Actually the above statement isn’t 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 true 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. In this lecture, we will study how this paradox relates to computation. Before doing so, let us consider another manifestation of this 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
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This note was uploaded on 09/06/2009 for the course CS 70 taught by Professor Papadimitrou during the Spring '08 term at University of California, Berkeley.

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n22 - CS 70-2 Spring 2009 Discrete Mathematics and...

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