Proof that there exist no finite axiomatic system with Compactness Theorem Say we would like to prove that the class of all infinite groups $(G, \circ, e)$ is not finitely axiomatizable by making use of the compactness theorem. We normally prove this by contradiction since we know the axiomatic system is defined by the union of $\Phi_{group}$ (the standard FO axiomatic system for groups) and $\Phi_{\infty}$ (in FO logic, there exist $x_1$ to $x_n$ and they are all pairwise non-identical and that for all $n \in \mathbb{N}$).
Assuming there exist a $\Phi_{infGrp} \subseteq FO(\tau)$ that finitely axiomatizes this class. Therefore, we can find a $\psi \in FO(\tau)$ that axiomatizes this class too, since the axiomatic system is finite.
This implies that $\Phi_{infGrp} \cup \{\neg\psi\}$ is not satisfiable. We then go on to find a arbitrary, finite subset  $\Phi_0 \subseteq \Phi_{infGrp} \cup \{\neg\psi\}$, and show that we can find a structure that is a model of this subset by making the said structure to have more elements than the formula with a largest $m \in \mathbb{N}$ in $\Phi_0$. We then finally say that since structure $\mathbb{A} \models \Phi_0$, by virtue of compactness theory, $\Phi_{infGrp} \cup \{\neg\psi\}$ is satsifiable.
The question is... how can $\Phi_{infGrp} \cup \{\neg\psi\}$ be satisfiable when $\neg\psi$ contains statements that negate the properties of a group. Wouldn't every structure (that is a group) not be a model for $\Phi_{infGrp} \cup \{\neg\psi\}$?

There is nothing in $\lnot\psi$ that needs to negate the properties of a group. Assume finite axiomatizability. So we can take the axioms to be $\alpha_1,\dots,\alpha_m$, the usual axioms of group theory, plus additional special axioms $\beta_1,\dots,\beta_n$, which in conjunction with the group axioms force the group to be infinite.  Then 
$\psi$ is the sentence
$$\alpha_1\land\cdots\land \alpha_m \land \beta_1\land\dots\land \beta_n.$$
The sentence $\lnot \psi$ is then the sentence
$$\lnot\alpha_1\lor\cdots\lor \lnot\alpha_m \lor \lnot\beta_1\lor\dots\lor \lnot\beta_n.$$
This is a disjunction. No group property needs to fail. 
