Peano and Osgood theorems via effective infinitesimals

We provide choiceless proofs using infinitesimals of the global versions of Peano’s existence theorem and Osgood’s theorem on maximal solutions. We characterize all solutions in terms of infinitesimal perturbations. Our proofs are more effective than traditional non-infinitesimal proofs found in the literature. The background logical structure is the internal set theory SPOT, conservative over ZF.