Keep pulling the thread on Joel David Hamkins.
Georg Cantor's major result was proving that the set of all real numbers is an uncountable set, meaning it represents a larger infinity than the countable set of natural numbers.
Cantor's diagonalization idea has proven to be an extremely fruitful proof method, forming the abstract basis for major results in mathematical logic such as Russell's paradox and the Halting problem.
The work of Kurt Gödel and Paul Cohen established that the Axiom of Choice will never be the source of an inconsistency in set theory.
Gödel's incompleteness theorems served as a decisive refutation of both primary goals of David Hilbert's program for formalizing mathematics.
Gödel's first incompleteness theorem states that any computably axiomatizable and consistent theory that includes a certain amount of arithmetic will be incomplete, meaning there will be statements it can neither prove nor refute.
Gödel's second incompleteness theorem states that no sufficiently strong and consistent theory can ever prove its own consistency.
The halting problem, which asks whether a given computer program will ever halt, is computably undecidable, as proven by Alan Turing.
In 1938, Kurt Gödel proved that the continuum hypothesis is consistent with the ZFC axioms of set theory by constructing a model of set theory called the constructible universe (L) where it holds true.
In 1963, Paul Cohen proved that the negation of the continuum hypothesis is also consistent with ZFC by inventing the method of forcing, thereby establishing the hypothesis's independence from ZFC.
The continuum hypothesis is known to be independent of all known large cardinal axioms, meaning these stronger axioms also cannot settle the question.
The problem of determining whether a specific cell in a given Conway's Game of Life configuration will ever become "alive" is computably undecidable and is equivalent to the halting problem.
Rice's theorem in computability theory establishes that all non-trivial semantic properties of programs are undecidable.