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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.

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  • $\begingroup$ Has $\sqrt{2}$ been proven irrational yet? If so, you may claim that you have no evidence for its irrationality yet either. $\endgroup$ – Alfred Yerger Jul 21 '18 at 16:20
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    $\begingroup$ 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. $\endgroup$ – lulu Jul 21 '18 at 16:22
  • $\begingroup$ 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. $\endgroup$ – DanielWainfleet Jul 30 '18 at 2:15
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At present, we have no evidence that there are any irrational numbers.

At the present time, there was no evidence for irrational numbers. However two pages later when you have the completeness axiom, THEN you get the existence of irrational numbers.

At present refers to the place in the text.

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  • $\begingroup$ Are you suggesting there is no difference between showing evidence and proving existence of irrational numbers? $\endgroup$ – user1068636 Jul 21 '18 at 16:19
  • $\begingroup$ @user1068636 you're wildly misinterpreting the passages. He's being a bit cheeky. "We have no evidence that there are any irrational numbers" is supposed to be read as, basically a joke. $\endgroup$ – user223391 Jul 21 '18 at 16:22
  • $\begingroup$ @user1068636 The point is without the completeness axiom you can't prove the existence of irrational numbers. That's the whole point. "showing evidence" is not a formal state and in context, is basically a joke $\endgroup$ – user223391 Jul 21 '18 at 16:25
  • $\begingroup$ I was actually thinking he might be referring to the sciences (i.e. is there any evidence of the square root of 2 in physics/chem etc.?) $\endgroup$ – user1068636 Jul 21 '18 at 16:26
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    $\begingroup$ @user1068636 No it has nothing to do with that $\endgroup$ – user223391 Jul 21 '18 at 16:26

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