Provide an interpretation where $∀_{x}∃_{y}G(x,y) ∧ ¬∃_{z}G(z,z)$ holds.
G(x, y) = x "greater than" y
This gives us the meaning that for all numbers, there exists a number where x is greater than y. However, there is no number that is greater than itself.
M = {U = Natural numbers, G(x, y) = 'x is greater than y'}"
Is this correct?
Your interpretation (in which the domain is the set of natural numbers $\{1,2,3,...\}$) is incorrect. To see why, first note the formula $\forall x \exists y G(x,y)$ is interpreted as follows: "for every $x$, there exists at least one $y$ such that $x > y$." In other words, there must always be an element $y$ that is less than every element $x$ in the domain. Now, let $x=1$. In that case, there is no natural number $y$ such that $x > y$ because $x=1$ is already the smallest natural number. As a result, the formula $\forall x \exists y G(x,y) \wedge \neg \exists z G(z,z)$ does not hold because the first conjunct $\forall x \exists y G(x,y)$ is false.
Instead, let your domain be the set of integers $\{...-2,-1,0,1,2,...\}$. Then, there will always be an integer $y$ smaller than every integer $x$ because $x-1$ always exists in the domain. It will also be the case that every integer cannot be greater than itself. Hence, both conjuncts of the formula $\forall x \exists y G(x,y) \wedge \neg \exists z G(z,z)$ will always be true.