TITLE: Differential equations and axiom of choice
QUESTION [15 upvotes]: In the most general context, the Picard-Lindelöf theorem (aka Cauchy-Lipschitz in French) asserts the existence of a maximal solution for $\dot{x}(t) = f(t,x(t))$, i.e. of a solution $x(t)$ defined on a interval $I$ such that there exist no other solution whose restriction  to $I$ coincide with $x$. The usual proofs of this (when $f$ is such that there is no local unicity) use Zorn's lemma, or some other weaker form of choice. But is this result actually not provable in ZF?

REPLY [11 votes]: 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 $t\in\mathbb{R}$
$$x ^ * (t):=\sup\big\{x(t)\, : x\in C^1(\mathrm{co}(t_0,t),\, \mathbb{R}),\, \mathrm{graph}(x)\subset\Omega, x(t_0)=x_0, \dot x(s)=f(s,x(s))  \big\}\, ,$$
(where of course $\sup\emptyset=-\infty$). Define
$\mathrm{dom}(x ^  *)$ to be the connected component of $t_0$ in the set $\{ t: x ^ *(t) \in\mathbb{R} \}$. Then, $x ^ *$ is a solution of the ODE with IVC $x(t_0)=x_0$, maximally defined on the interval $\mathrm{dom}(x ^  *)$.