If the order in a set doesn’t matter, can we change order of, say, $\Bbb{N}$? I’m given to understand that the order of the elements of a set doesn’t matter. So can I change the order of the set of natural numbers or any set of numbers ( $\mathbb{W,Z,Q,R}$ for that matter) as follows?
$$ \mathbb{N} = \{ 1,2,5,4,3,\cdots \} $$
$$ \mathbb{N}= \{3,5,\cdots,78,1,9\} $$ 
If yes, is this is really problematic? If not, then why not?
PS: I know this is a silly question, but this has been in the back of my mind for a really long time.

If set extension definition would required set to be ordered, it would imply that you wouldn’t be able to define in extension a set without an order.
Fortunately, you can define sets that are not ordered! Moreover, you may define several orders on a set. Which one would you use for its definition ?
