Well-ordering principle and theorem

Question

Could somebody clearly explain the difference between the well-ordering principle and the well-ordering theorem? Is one of these related to the Principle of Mathematical Induction, and the other to the Axiom of Choice? Thanks.

Answer 1

The "well-ordering theorem" is the statement that for any set $X$, there is a relation $<$ on $X$ which is a well-ordering. This statement is equivalent to the axiom of choice.

The "well-ordering principle" has (at least) two different meanings. The first meaning is just another name for the well-ordering theorem. The second meaning is the statement that the usual relation $<$ on the set $\mathbb{N}$ is a well-ordering. This statement is equivalent to the statement that ordinary induction on the natural numbers works.