These are terms that are significant to the history of this project and of formal systems. The five axioms that characterize natural numbers, addition, and multiplication. I dunno how old. Euclid's 5 Axioms of plane geometry Lots have folks have tried to express one of these in terms of the other four. Let A be the set of all sets that are not members of themselves.
Every formal systems is either incomplete or inconsistent.
The halting problem There is no algorithm that will decide for an arbitrary computer program whether that program halts or computes forever. Bourbaki Another effort to codify mathematics.