Applications in Other Branches of Mathematics

In this chapter, we shall give applications of set theory to other branches of mathematics. We shall assume that the reader is familiar with the subjects and give only preliminary definitions needed to understand the statements and proofs of the applications that we shall be presenting. Let I be an infinite set and for each \(i\in I\) let \(a_i\) be a non-negative real number. We define \(\sum _i a_i = \sup \{\sum _{i\in F}a_i: F\subset I ~\text{ finite }\}.\) Note that this extends the usual definition of the sum \(\sum _{j\in J}a_j\) when each \(a_j\ge 0\) and J countable to uncountable sum of non-negative real numbers.