Here’s something I didn’t know. There exists nonlinear (but algebraic) ordinary differential equations such that solutions to that differential equation are dense in the space of continuous functions. These are known as universal differential equations. An explicit construction of one is given in this preprint by Keith Briggs. If I understand the construction correctly, the trick seems to be that the nonlinearity gives you branch points where you have a choice for the direction in the solution. This allows you to paste together solutions in enough ways that you can achieve density.

This thread at Math Overflow has the feel of advanced alien technology. Forcing is a technique for constructing models of set theory where various hypotheses fail. For example, forcing can be used to construct a model of set theory where the continuum hypothesis is violated.

There are some statements whose value cannot be affected by forcing. These statements are known as absolute. Forcing is useless for establishing such a statement is independent, but this can be a virtue. If you can create a model using forcing such that you can prove that an absolute statement is true in that model, then it must already be true in the universe of ordinary sets. The thread gives several specific examples of theorems you can prove this way.