I am trying to understand why is the well ordering principle stated as an axiom of integers.
In the process, I found a "proof" of the principle (which is obviously wrong) and want to understand where the error is.
Let $S$ be a subset of $\mathbb{N}$, let $n \in S$. Then we have :
- Either $n$ is less than every other element of $S$ and then $S$ has a least element.
- Either the above is false, then take $m$ to be least element of the finite set $S\cap [0\ldots n]$ which is necessarily non empty (because if contains at least $n$ itself) and then $m$ will be less than any element of $S$.
Where is the error? Is it the use of axiom of choice when I take "let $n \in S$"? In that case is it true that axiom of choice implies well ordering principle of integers? (I know axiom of choice implies well ordering theorem, but here i am talking about well ordering principle of integers with natural order).
May be the error is assuming that every finite subset of $\mathbb{N}$ has a least element? Thanks for help.