Analysis Of Richard's Paradox Of The Liar

The third chapter discusses three famous paradoxes to illustrate how the mind works the concepts introduced previously. The three paradoxes are: Berry paradox, Richard’s paradox, and the paradox of the Liar. All three paradoxes explore the limits of human languages and how that affects our understanding of mathematical concepts. The Berry paradox is particularly concerned with numbers and their descriptions; there are numbers that are cannot even be named for example. I liked how the author described the relationship of the Berry paradox to computation. He avoided turing machines, which might have been unnecessary overhead for the general audience and instead used an IBM typewriter. The rest of the section describes the idea of uncomputable numbers in a similar …show more content…
There is some machine M that can give the nth digit of some real number given some name or description of the real number. Then M is closed if whenever M has a name for some real number s, then M also has a name for real numbers that have simple definitions in terms of s. This leads to a contradiction where M cannot name some diagonal number (constructed using the table used in lecture to prove that the reals are uncountably infinite) but if M has a name for T and M is closed, it must name this diagonal number too. This, similar to the Berry paradox, demonstrates a finite scheme cannot “capture the essence of how one connects the real and the ideal.” The last paradox is one most of us are familiar with. It is sentences that seem to contradict themselves, such as “This sentence is false.” The remainder of the chapter discusses this paradox in detail, but at the end it reaches a similar conclusion as the other two paradoxes: there is an inherent limitation in our language, semantics, and representation. The first paradox shows this through the natural numbers, the second paradox through the real numbers, and the last through truth