If the order in a set doesn’t matter, can we change order of, say, $\Bbb{N}$?
It's true that sets are not ordered. As to whether you can 'change' the order, you cannot change something that is not there.
However you can define any ordering on them you want. For instance, we can order the naturals the usual way $$0,1,2,3,\ldots$$ or we can define an ordering where all the even numbers come first in their usual order, then the odd numbers $$ 0,2,4,6,\ldots, 1,3,5,7,\ldots.$$ There are many, many possibilities.
Also, the ordering needs to be defined unambiguously so we know exactly the order relationship between any two elements. For instance, I don't know what you mean when you write $$ \{3,5,...,78,1,9\}.$$ It's clear that you mean $9$ comes last (it is not a problem for an ordering to have a greatest element, even though in the two orderings I gave above, there was no greatest element), but I have no idea where $2$ goes in this ordering. If you just wrote this out of the blue, I wouldn't even be able to tell it was an ordering of the whole set of natural numbers and not just a subset.
Edit
Henning mentions an example in the comments that I think deserves mention in the answer, to reinforce the fact that there are many possibilities. Any enumeration of a countably infinite ordered set induces an order on the natural numbers. So, from the usual ordering of the rationals and an enumeration of the rationals, we get an ordering on the natural numbers that is dense, i.e. between any two numbers lies infinitely many others. We can’t even try to communicate this ordering as a list with some ellipses.
If you have the set $\{0, 1, 5, 3\}$, you can write it as $\{0, 3, 1, 5\}$. These are the same set, they contain the same elements.
And this is not changing the order of elements in the same set, as there is no order of elements in the same set. You're talking about the same set, just using different words. If I say
- "My family contains me, my wife, and our daughter",
and then I say
- "My family contains our daughter, my wife, and me",
I didn't change the order of members in my family, that doesn't even mean anything in this context! I just used different words to say the same thing.
But the crux of your question isn't about whether you can reorder the elements in a set, it really about what "..." means.
The "..." is a funny thing in mathematical texts. It's not really a mathematics concept at all, it's short of "dear reader, you know what I would write here if there was enough (possibly infinite) space and time, so let's just pretend I wrote it here".
When you write "$\{1, 2, 3, 4, ...\}$", the reader knows that if you had enough space, you'd write all positive integers there, so he knows you mean the set of positive integers. When you write "$\{1, 2, 5, 4, 3, ...\}$", the reader has no clue what you would write there. The problem isn't that you wrote the set in different order, the problem is that you weren't clear enough when you told the reader what you mean.
Again back to a (slightly clumsy) natural language example:
- "I remember the names of the 3rd US president, the 4th US president, the 5th US president, and so on and so on."
The reader probably realizes you mean you know the names of all US presidents, except perhaps for the first two.
- "I remember the names of the 30th US president, the 6th US president, the 17th US president, and so on and so on."
The reader has no clue what you mean.
In ZFC the two sets are the same set. I don't see why it is problematic.