In the textbook "Advanced Calculus" by Patrick Fitzpatrick, on page 7 it says:
A real number is called irrational if it is not rational. At present, we have no evidence that there are any irrational numbers.
Then later on on page 9 it references the Completeness Axiom:
At first glance, it is not at all apparent that the Completeness Axiom will help our development of mathematical analysis...the Completeness Axiom guarantees that there is a number, necessarily irrational, whose square equals 2.
These two quotes seem contradicting to me. How is it possible there is "no evidence" of irrational numbers on page 7, but then on page 9 the Completeness Axiom "guarantees that there is a number, necessarily irrational.."?
Am I missing something here?
Here is the Completeness Axiom:
Suppose that $S$ is a nonempty set of real numbers that is bounded above. Then, among the set of upper bounds for $S$ there is a smallest, or least, upper bound.