Why is a class of, say, finite groups $(G,\circ,e)$ not axiomatizable by FO logic (we use the compactness theorem to prove this statement) but a finite linear order $(A,<)$ on the other hand can be axiomatizable by FO logic. I still don't really get the idea behind that.
(Okay I merely assume that the latter would be axiomatizable because the exercise I got asked for a formula that can be satisfied by $(A,<)$. I'm guessing we could simply define the existence of a maximum and minimum element which implies finiteness of a set?)
I thought I would expand somewhat on my comment.
It is true that the class of all finite groups is not first order axiomatizable. However it is also true that any particular finite group $G$ is first order axiomatizable. To see this, say the order of $G$ is $n$, then we can write out a big sentence which says there exist $n$ different elements, and there are no more than $n$ elements, and then we can just describe the multiplication table for this group, this defines the group upto isomorphism.
Similarly collection of all finite linear orders is not first order axiomatizable, but any particular finite linear order is.