Recursive definition of positive real Number

Question

The following is a recursive definition of positive real numbers from book "Computer Theory" by I. Cohen.

1. 1 is in positive R
2. If x and y are in R, then so x+y, xy, and x/y

but the author said that

it does define some set, but it is not the set of positive real numbers

What does it mean as all the positive numbers are in the set defined by the above definition?

Answer 1

Those are all rational operations, so that set is not the positive real numbers because it doesn't include any positive irrational numbers (e.g. sqrt(2)).