I'm taking a discrete mathematics course right now and I can't quite understand what a well ordered set is. From what I do understand, a set is considered to be well ordered if it's a totally ordered set and all non empty subset has a smallest element.
So $(ℕ, \leq)$ is considered a well ordered set because it's a totally ordered set and $0$ is the smallest element. Isn't this true for the set $(ℕ, \geq)$ as well? From what I understand, it's a totally ordered set and $0$ would be the smallest element but the notes from the professor says that it's not.
Thanks
(edit : typo)
Best Answer
You're mixing up the terminology. "Smallest" with respect to $\geq$ is the "Largest" with respect to $\leq.$
In other words, saying $(\mathbb{N}, \geq)$ is well-ordered would imply that $\mathbb{N}$ has a largest element with respect to the usual order. Does it?