semantic consequence in first order logic I'm asked to show that Fx does not semantically entail AxFx. However in the preceding paragraph the author tells me Fx is true in a model iff AxFx is true in that model. So how am I supposed to provide a model in which Fx is true and AxFx is false?
Here the definition of semantic consequence is from Ted Sider's Logic for Philosophy:
PHI is a semantic consequence of a set of well formed formulas GAMMA if and only if for every model M and every variable assignment g for M, if each member of GAMMA is true, then PHI is true.
 A: It depends on the details of the definitions of "true in a model" and "semantical entailment". 
In :


*

*Herbert Enderton, A Mathematical Introduction to Logic (2nd ed - 2001), page 88, we have that :



DEFINITION. Let $\Gamma$  be a set of wffs, $\varphi$ a wff. Then $\Gamma$ logically implies $\varphi$, written $\Gamma \vDash \varphi$, iff for every structure $\mathfrak A$ for the language and every function $s : Var → |\mathfrak A|$ such that $\mathfrak A$ satisfies every member of $\Gamma$ with $s$, $\mathfrak A$ also satisfies $\varphi$ with $s$.

According to this definition, it is not true in general that $\varphi(x) \vDash \forall x\, \varphi(x)$.
Consider as $\varphi$ the formula of first-order arithmetic : $(x=0)$. It is clear that with an $s$ such that $s(x)=0$ :

$\mathbb N \vDash (x=0)[s]$;

but clearly $\forall x\,(x=0)$ is not satisfied by $s$ in $\mathbb N$, and thus :


$(x=0) \nvDash \forall x\,(x=0)$.



As you can see in :


*

*Theodore Sider, Logic for Philosophy (2009), page 115


this is the same as :

Definition of semantic consequence : $\phi$ is a PC-semantic consequence of set $\Gamma$ of wffs  (“$\Gamma \vDash_{PC} \phi$”) iff for every PC-model $\mathscr M$ and every variable assignment $g$  for $\mathscr M$, if $V_{\mathscr M,g} (\gamma) = 1$ for each $\gamma \in \Gamma$, then $V_{\mathscr M,g} (\phi) = 1$.

See page 114 :

Definition of truth in a model: $\phi$ is true in PC-model $\mathscr M$ iff $V_{\mathscr M,g} (\phi) = 1$, for each variable assignment $g$ for $\mathscr M$.

Thus, we have that $\phi$ is true in $\mathscr M$ iff $\forall x \, \phi$ is.
If we apply this definition to my example above, we have that $(x=0)$ is not true in $\mathbb N$ simply because it is not true that $V_{\mathbb N,g} (x=0) = 1$, for each variable assignment $g$ : it is enough to consider a $g$ such that $g(x)=1$.
And thus, we have that both $(x=0)$ and $\forall x\,(x=0)$ are false in $\mathbb N$.
