Thread: Prove 1 is the least element in the positive integers

I did a proof, it sounds right to me, but I want to make sure.

Prove: 1 is the smallest element in Z+ (the positive integers)

Suppose not, then there exists some integer such that

Let . By our assumption that , . Thus S is not the empty set and by the Well Ordering Principle there exists a smallest element, call it . Since it follows that .

Thus, which implies that . It follows that . Thus . But , a contradiction since is the smallest element in S. Thus 1 must be the smallest element in the positive integers.

Is this right?

Originally Posted by superevilcube

I did a proof, it sounds right to me, but I want to make sure.

Prove: 1 is the smallest element in Z+ (the positive integers)

Suppose not, then there exists some integer such that

Let . By our assumption that , . Thus S is not the empty set and by the Well Ordering Principle there exists a smallest element, call it . Since it follows that .

Thus, which implies that . It follows that . Thus . But , a contradiction since is the smallest element in S. Thus 1 must be the smallest element in the positive integers.

Is this right?

Using the well ordering principal you can prove what you want to prove in the following way:


The principal says that:

For all sets S, S N and S=/= Φ ( S being non empty) S has a least element.

Since the above holds for all non empty subsets of N and since N is non empty and a subset of itself then N has a least member.


Alternatively we can prove the above by induction in the following way:


for n=1 we have .............1>=1 and the theorem is true for n=1


suppose now the theorem true for n=k and k>=1 .And we need to prove that : k+1>= 1.


But k>=1 ====> k+1>= 1+1 >1 +0, since 1>0 and so


................k+1>1 =====> k+1>=1 and the theorem is proved

I think what poutsos is trying to say is that the well ordering principle is a consequence of 1 being the smallest element of . He may or might not be right, depending on what axioms you are using. In the construction I learnt, 1 was defined as the smallest positive integer.

Your proof is entirely sound if you are allowed the well ordering principle.