Set theory
Cantor's theory of infinite sets — and the standard foundational language modern mathematics is written in.
Founded: 1874 — Cantor's first paper on the uncountability of the reals
The claim
No credible dismissal found. The famous “set theory is a disease” line usually pinned on Poincaré is documented as apocryphal (Jeremy Gray, The Mathematical Intelligencer, 1991). Kronecker's attacks on Cantor (“corrupter of youth”) target set theory's legitimacy — whether actual infinities exist at all — not its utility.
What happened
No major real-world application is well-documented today. Split by what “application” means. Elementary set theory is enabling infrastructure for essentially all of modern mathematics: functions are sets of ordered pairs, spaces are sets with structure, probability spaces are sets equipped with σ-algebras and measures. That's a real transmission mechanism, just an indirect one — it doesn't put set-theoretic notation into a GPS receiver, it puts a foundational language under the mathematician who writes the receiver's equations. Application of mathematics is not the same thing as its notation appearing in the final result. The narrower, genuinely open question is whether the distinctive machinery of research set theory — forcing, measurable and large cardinals, independence results, inner models, none of which ordinary mathematics actually needs — has any application outside mathematics. There, the answer is still no. Descriptive set theory comes closest (Borel equivalence relations have been used to establish measurable equilibria in Bayesian and stochastic games), but it sits close enough to analysis and topology that it isn't a clean vindication of forcing or large-cardinal set theory specifically. Turing's 1936 reuse of Cantor's diagonalization is a reuse of a proof technique, not an application of transfinite set theory itself.
Verdict: No external application documented.
Cantor began in 1874; Cohen's forcing is from 1963. This may simply be a field sitting inside its own waiting period, the same one every other field on this page eventually left.
Confidence: medium.