$$2^{\aleph_0} = \aleph_1$$
The Continuum Hypothesis is an example of an undecidable statement par excellence. It is an example of a problem that is:
- natural;
- historied -- it was first asked by Cantor himself;
- celebrated -- it was Hilbert's first of his famous 23 problems; and
- undecidable in a very deep way -- it's independent of all current large cardinal axioms, and in fact something deeper than this can be said, but that is currently beyond my understanding.
Moreover, the desire to "solve" CH was the main motivation, or at least one of the main motivations, for:
- Godel's description of the Constructible Universe, which has already been listed as a fundamental example in logic and foundations, and the beginnings of inner model theory; and
- Cohen's invention of the method of forcing which is now indispensable and ubiquitous in modern set theory, and also won Cohen a Fields medal.
