TITLE: What is a natural number?
QUESTION [7 upvotes]: According to page 25 of the book A First Course in Real Analysis, an inductive set is a set of real numbers such that $0$ is in the set and for every real number $x$ in the set, $x + 1$ is also in the set and a natural number is a real number that every inductive set contains. The problem with that definition is that it is circular because the real numbers are constructed from the natural numbers.

REPLY [5 votes]: You have to start somewhere. 
And in mathematics, you have to start from some axioms.
If you choose to start from the axioms for the real numbers, then using them you define the natural numbers exactly as you quoted, and using that that definition you can prove that the Peano axioms are true.
If instead you choose to start from the Peano axioms for the natural numbers, then using them you can define the real numbers, and using that definition you can prove that the axioms for the real numbers are true.
Either one of those provides an answer to your question. You may not like that answer, but that's how things go. What you cannot do is to avoid choosing your axioms.
Nonetheless, perhaps there is still a way to break this vicious loop?
Well, actually, there is, by making a different choice of axioms. You can, instead, choose the basic axioms for set theory, also known as the Zermelo-Frankel axioms or ZF for short. 
If you choose to start from the ZF axioms, then using them you can define the natural numbers using the Von Neumann definition. Using that definition, you can then prove that the Peano axioms are true. And then, as already said, you can use the Peano axioms (which in this scheme are now theorems instead of axioms) to define the real numbers and use that definition to prove that the axioms for the real numbers are true.