Analyze the following popularization of the Russell paradox: Every municipality of a certain country must have a mayor, and no 2 municipalities may have the same mayor.  Some mayors do not reside in the municipalities they govern.  A law is passed compelling nonresident mayors to reside by themselves in a certain special area A. There are so many nonresident mayors that A is proclaimed a municipality.  Where shall the mayor of A reside? (If the mayor resides in A, then ... .  If the mayor does not reside in A, then ... .)

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This is a variant on what is normally framed as the paradox of the village barber who shaves all men who do not shave themselves. The paradox has historically been used as an argument for the incompleteness of set theory. One of the more interesting treatments of this paradox, regarded by some as a definitive solution, can be found in an essay by Wilfred van Orman Quine called "The Ways of Paradox." He resolves it by stating that there can be no such barber (or mayor). This is not just a trivial solution because it challenges the correspondence theory of meaning by pointing out that there are sentences that are intelligible without corresponding to some external state of affairs, and thus eliminating the need for recasting according to Russell's theory of description.

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