Points, Lines, and the Structure of \({ \mathbb {R}}\)

In the previous chapter we saw how a large portion of mathematics can be formalized in first-order logic. The very fact that the construction of the classical number structures can be formalized this way makes first-order logic relevant, but is it necessary? For centuries mathematics has been developing successfully without much attention paid to formal rigor, and it is still practiced this way. When intuitions don’t fail us, there is no need for excessive formalism, but what happens when they do? Intuition can be misleading, especially when infinity is involved. In this chapter, we will see how seemingly innocuous assumptions about actually infinite sets lead to consequences that are not easy to accept. Then, we will go back to our discussion of a formal approach that will help to make some sense out of it.