Given a conjecture, the best thing is to prove it. The second best thing is to disprove it. The third best thing is to prove that it is not possible to disprove it, since it will tell you not to waste your time trying to disprove it. That's what Gödel did for the Continuum Hypothesis.
There is an old maxim that says that two empires that are too large will collapse. The analog in set theory is that two different theories that are too powerful must necessarily contradict each other.
Cardinal Arithmetics is much older than Number Theory. People used to exchange things way before there were numbers. Expressing numbers like 762 is already a sign of a very advanced civilization.
Number theorists say that number theory is too complicated, so let's pretend that there is only one prime number, and then let's combine all these results. Surprisingly, sometimes it works.
