Good Math

17Axiomatic Set Theory: Keep the Good, Dump the Bad

In the last chapter, we saw the basics of naive set theory the way that Cantor defined it. Naive set theory seems wonderful at first because of its brilliant combination of simplicity and depth. Unfortunately, that simplicity comes at great cost: it allows you to create logically inconsistent self-referential sets.

Happily, the great mathematicians of the early twentieth century weren’t willing to give up on set theory. The devotion of mathematicians to preserving set theory is best summed up by the words of one of the most brilliant mathematicians in the history of math, David Hilbert (1862--1943): “No one shall expel us from the Paradise that Cantor has created.” This devotion led to an ...