Philosophy of Mathematics

1. It has been explained that to show that a set of items is countable, all you have to do is use the method of naming (digits), that is, name all members of the set, giving each member a unique name, where a name is a finite string of symbols drawn from a finite alphabet. Explain why the method of naming works.
2. Using the method of naming, show that the following sets are countable. In each case explain what the symbols are, and how each member of the set is uniquely named. (The integers are the numbers 0, 1, 2, 3, ...)
a. the set of all fractions whose numerators and denominators are integers
b. the set of all triples of integers, where a triple < j, k, l> is a sequence of three integers j, k, l and the order matters
c. the set of all sentences in the English language
d. the set of all possible books, where a book is a sequence of sentences
e. the set of all possible libraries, where a library is any collection of books
3. Using Cantor's diagonal method, (see section 33), show that the real numbers are not countable. Why does the method of naming NOT show that the reals are countable?