At least for scalar equations $\dot x(t)=f(t,x(t))$, that is with a nonlinearity $f\in C^0(\Omega,\mathbb{R})$, defined on an open set $\Omega\subset \mathbb{R}^2$, the Zorn's lemma is not necessary: the order structure of $\mathbb{R}$ allows to select a preferred solution (actually, two)
Any IVP admits an upper and a lower solution, whose domains are maximal intervals. For $(t_0,x_0)\in\Omega$, define, for $t\in\mathbb{R}$ $$x ^ * (t):=\sup\{x(t)\ : x\in C^1(\mathrm{co}(t_0,t),\ \mathbb{R}),\ \mathrm{graph}(x)\subset\Omega, x(t_0)=x_0, \dot x(s)=f(s,x(s)) \}\ ,$$ (where of course $\sup\emptyset=-\infty$). Define $\mathrm{dom}(x ^ *)$ to be the connected component of $t_0$ in the set $\{ t: x ^ *(t) > -\infty\}$. \in\mathbb{R} \}$. Then,$x ^ *$is a solution of the ODE with IVC$x(t_0)=x_0$, maximally defined on the interval$\mathrm{dom}(x ^ *)$. 1 At least for scalar equations$\dot x(t)=f(t,x(t))$, that is with a nonlinearity$f\in C^0(\Omega,\mathbb{R})$, defined on an open set$\Omega\subset \mathbb{R}^2$, the Zorn's lemma is not necessary: the order structure of$\mathbb{R}$allows to select a preferred solution (actually, two) Any IVP admits an upper and a lower solution, whose domains are maximal intervals. For$(t_0,x_0)\in\Omega$, define, for$t\in\mathbb{R}$$$x ^ * (t):=\sup\{x(t)\ : x\in C^1(\mathrm{co}(t_0,t),\ \mathbb{R}),\ \mathrm{graph}(x)\subset\Omega, x(t_0)=x_0, \dot x(s)=f(s,x(s)) \}\ ,$$ (where of course$\sup\emptyset=-\infty$). Define$\mathrm{dom}(x ^ *)$to be the connected component of$t_0$in the set$\{ t: x ^ *(t) > -\infty\}$. Then,$x ^ *$is a solution of the ODE with IVC$x(t_0)=x_0$, maximally defined on the interval$\mathrm{dom}(x ^ *)\$.