# What is the difference between showing evidence of irrational numbers and proving their existence?

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.

• Has $\sqrt{2}$ been proven irrational yet? If so, you may claim that you have no evidence for its irrationality yet either. – Alfred Yerger Jul 21 '18 at 16:20
• This is just semantics. The author is saying that a priori we don't know that there are irrational numbers in the reals. After some thinking about the subject, that changes. It's not meant to be a formal statement. – lulu Jul 21 '18 at 16:22
• I think what the author means on page 7 is that what he has presented in the book up to that point has not included any demonstration that irrationals exist. – DanielWainfleet Jul 30 '18 at 2:15