What are the problems and theorems that are derived from it?
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There are actually two incompleteness theorems. Both apply to formal systems, which are logical systems where specific rules are applied one by one to generate proofs. Mathematicians use formal systems as a way to formalize certain kinds of logical thought, and almost all mathematics can be reduced to a formal system. The incompleteness theorems prove that there are limits on any formal system, and therefore limits on our ability to describe math.
The first theorem more or less says that any formal system with enough complexity is either inconsistent or incomplete. “consistent” means that there are no contradictions in the system’s rules. In mathematics, even a single contradiction ruins the whole system, so consistency is desirable. “complete” means that any statement in the system can be either proved or disproved by some clever arrangement of the system. An incomplete system then has statements that are neither true nor false; the rules of the system just don’t say anything about them.
This theorem is “bad news” because it means there’s no logical system we can set up that describes “all of math.” No matter what rules we decide we want to use, there will be some “facts” that those rules can’t be used to reason about, unless the rules contradict each other, which we really don’t want. We could decide whether any of those statements “should” be true or false, and then add another rule to the system saying so, but the decision about whether they “should” be true would be a human judgement call, not a mathematical fact, and there may be people who support using both.
The second theorem says that any consistent formal system with enough complexity can’t be used to prove its own consistency. Effectively, if your logical system works, you can’t *prove* that it works, unless you use some other logical system to prove it. But then you can’t prove that *that* logical system works without even another system, and so on…
This is “bad news” because it means we’ll technically never know if the logic we’re basing our mathematics on is truly consistent, or if there’s some contradiction lurking somewhere that would ruin the whole thing. This is considered extremely unlikely for the most commonly used systems, but there’s no way to say for certain.
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