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Feldmann Denis
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Differential equations and axiom of choice

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?

Feldmann Denis
  • 3.6k
  • 1
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  • 36