# Is the Least Integer Principle an axiom?

I'm really sorry if this question has been asked before, I looked but couldn't find anything.

I'm going through an elementary number theory book and in the first chapter it introduces the least integer principle. There is no proof accompanied with it, it just states: "a nonempty set of integers that is bounded below contains a smallest element."

Is the reason the book doesn't give a proof because we take this as an axiom? Or is the proof so trivial that it need not be written?

Thanks!

• It depends. It need not be an axiom, but sometimes it is. – Git Gud Aug 14 '14 at 18:08
• It's an equivalent way to express the principle of induction on $\,\Bbb N,\,$ look up well-ordered. – Bill Dubuque Aug 14 '14 at 18:15
• Given the Peano construction of the natural numbers, you can prove the well-ordering property using induction. – RRL Aug 14 '14 at 18:17
• If we formalize set theory (the most commonly used formalization is ZFC) then the integers can be defined, and the Least Integer Principle is a theorem. – André Nicolas Aug 14 '14 at 18:28

The thing is one of these axioms must be either the "Well-Ordering Principle" or the "Principle of Mathematical Induction" which we use to inductively construct the whole set $\Bbb N$. So at this level one of these two principles must be accepted as part of the definition of the set $\Bbb N$. If you accept Mathematical Induction you can prove the Well Ordering Principle as a theorem. If you accept the Well-Ordering Principle you can use it to prove the Principle of Mathematical Induction. That is what people say when the two conditions are equivalent. Note that both conditions claim to be a "Principle" and not a theorem.