In 1940 Gödel proved the consistency of the continuum hypothesis with the Zermelo-Fraenkel axioms of set theory, by introducing the constructible universe $L$ and subsequently founding the subfield of set theory now known as inner model theory. In 1963, Paul Cohen introduced the method of forcing, which he used to show the continuum hypothesis is independent from the Zermelo-Fraenkel axioms, resolving Hilbert's first problem. However, forcing can also be used to give an alternative proof of the consistency of the continuum hypothesis, giving two proofs.

In 1940 Gödel proved the consistency of the continuum hypothesis with the Zermelo-Fraenkel axioms of set theory, by introducing the constructible universe $L$ and subsequently founding the subfield of set theory now known as inner model theory. In 1963, Paul Cohen introduced the method of forcing, which he used to show the continuum hypothesis is independent from the Zermelo-Fraenkel axioms, resolving Hilbert's first problem. However, forcing can also be used to give an alternative proof of the consistency of the continuum hypothesis, giving two proofs.

The computable functions provably total in Peano arithmetic are exactly the $\varepsilon_0$-recursive functions. According to Buss's "The Witness Function Method and Provably Recursive Functions of Peano Arithmetic", this has been proven at least using Gentzen-style assignment of ordinals to proofs, Gödel's Dialectica interpretation, and model-theoretic methods.

In 1940 Gödel proved the consistency of the continuum hypothesis with the Zermelo-Fraenkel axioms of set theory, by introducing the constructible universe $L$ and subsequently founding the subfield of set theory now known as inner model theory. In 1963, Paul Cohen introduced the method of forcing, which he used to show the continuum hypothesis is independent from the Zermelo-Fraenkel axioms, resolving Hilbert's first problem. However, forcing can also be used to give an alternative proof of the consistency of the continuum hypothesis, giving two proofs.

In 1940 Gödel proved the consistency of the continuum hypothesis with the Zermelo-Fraenkel axioms of set theory, by introducing the constructible universe $L$ and subsequently founding the subfield of set theory now known as inner model theory. In 1963, Paul Cohen introduced the method of forcing, which he used to show the continuum hypothesis is independent from the Zermelo-Fraenkel axioms, resolving Hilbert's first problem. However, forcing can also be used to give an alternative proof of the consistency of the continuum hypothesis, giving two proofs.