I'm really struggeling with a logic task and I need a hint how to solve these kind of Problems.
Q: Which of the ZFC Axioms are valid on the Structures:
(1) ($Z$,<)
(2) (R,<)
I really dont see how any of the ZFC Axioms are not valid because R, Z are sets...
The ZFC axioms do not describe sets - rather, they (purport to) describe how the universe of sets behaves with respect to "$\in$". That is, "model of ZFC" doesn't mean "thing which is a set" but "structure which looks like the universe of all sets."
To see how this plays out, let's take a couple of the more concrete ZFC axioms and look at how they behave in general structures in the appropriate language (= one binary relation $E$):
Pairing: $\forall x,y\exists z\forall u(uEz\leftrightarrow (u=x\vee u=y))$. When we look at the structure $(\mathbb{R};<)$, this statement is clearly false: no real number is greater than only two real numbers!
Extensionality: $\forall x,y(\forall z(zEx\leftrightarrow zEy)\leftrightarrow x=y)$. This statement is clearly true in $(\mathbb{R};<)$, since any two real numbers which are greater than exactly the same things are equal.
Now you need to continue through the remaining axioms ...