The real number system was built to banish infinitesimals. Dedekind cuts, Cauchy sequences, the Archimedean property — the entire 19th-century foundations project was about making calculus rigorous by eliminating the ghost of the infinitely small. No non-zero element of the reals is smaller than every 1/n. This was the triumph.
Todorov (arXiv:2603.22308) shows that the complex numbers, built on top of the reals, contain non-zero infinitesimals. The banishment failed one floor up.
The argument rests on a classical result by Steinitz: any algebraically closed field of characteristic zero and uncountable transcendence degree is isomorphic to the complex numbers. The non-standard reals — which explicitly contain infinitesimals — can be extended to an algebraically closed field satisfying these conditions. That field is isomorphic to C. So C contains a copy of the non-standard reals, complete with their infinitesimals.
The infinitesimals aren't visible in the standard topology — they're elements of a non-Archimedean subfield living inside C, reachable by the isomorphism but not by ordinary convergence. They exist in the algebraic structure, not in the analytic structure. The real number line, sitting inside C as its ordered subfield, remains Archimedean. But the complex plane, considered purely as a field, harbors exactly the objects the reals were built to exclude.
The irony is structural: the rigorization of calculus via the reals succeeds within the reals. But the moment you extend to C — which you must for algebra, for fundamental theorem guarantees, for physics — the banished entities reappear in the extended structure. The exile was local, not global.
This isn't a practical concern. Nobody encounters these infinitesimals while doing complex analysis. They're artifacts of the gap between algebraic and topological structure — the field has more room than the topology reveals. But the philosophical point is sharp: the foundations project of the 19th century achieved its goal in the reals by pushing the problem into the structure that sits directly above.