Worked solution: Gödel and Cohen: the continuum hypothesis is independent of ZFC (1963)
Gödel's (Steps 3–4) shows the parallel-postulate-style picture is real in one direction: assuming never creates a contradiction with . Cohen's forcing (Steps 5–7) shows the same for the opposite direction: assuming never creates a contradiction either.
With both halves in hand, exactly like discovering both elliptic and hyperbolic geometries are consistent alongside flat Euclidean geometry, is confirmed to be a genuine fork in the road that the axioms simply leave open.
Combining Gödel's result () with Cohen's (): relative to the consistency of itself, both and are consistent theories. By the definition from Step 2, this is precisely what it means for to be independent of : neither nor is provable from alone.
Cohen received the Fields Medal in 1966 largely for this achievement — to date, forcing is the only technique ever to win its inventor a Fields Medal, reflecting how completely it reshaped set theory. Both halves were essential: Gödel's half alone (known since 1940) left CH's provability open for over twenty years, and it was only Cohen's second half in 1963 that completed the independence result Hilbert's first problem had been implicitly asking about.
Forcing did not stop with Cohen's original construction; it became the primary tool of modern set theory for building custom models with prescribed properties, which is exactly the technology the remaining step surveys.