Example of Set which possesses well ordering property other than Integers

Question

During Studying Elementary Number theory I had encountered in property called as well ordering property which tell every nonempty set of natural number has least element.
I had interested in is such property hold for any other set.If not then How to prove that. As I had checked one question A well-order on a uncountable set but I had not understood.
Any Help will be appreciated.

Answer 1

Let $N$ be a countable set. This means that there's a bijection $b\colon\mathbb{N}\longrightarrow N$. Define this order relation on $N$: $a\leqslant a'$ if and only if $b^{-1}(a)\leqslant b^{-1}(a')$. Then $(N,\leqslant)$ is a well-ordered set.

Answer 2

What you call "well ordering property" is probably the well ordering principle and is formulated explicitly for the set of positive integers.

"I had interested in is such property hold for any other set"

It can be projected, but not purely on sets. For that you need sets that are equipped with an ordering.

The axiom of choice is in the context of ZF equivalent with the statement that every set can be equipped with such an ordering, which is a so-called well-ordering.

Answer 3

Here's another example to think about. The underlying set is the natural numbers, but we redefine the ordering: if $m$ and $n$ are both odd or both even then $m "<" n$ just when $m < n$ in the usual sense, but every even number is (by definition) $"<"$ than every odd number. The ordering looks like this $$ 0,2,4, \ldots 1, 3, 5, \ldots $$