I’m trying to understand the incompleteness theorem and am having a rough time. The only part I understand is that math can’t prove everything. The rest I am having trouble.

Can someone explain it to me in simple layman terms?

Basically, Gödel figured out how to assign a unique number to every expression and proof in a formal logical system. That allowed him to construct a statement that effectively says, “I can’t be proved.” If the system is consistent, the statement is true—but it can’t be proved from within the system.

The cool part isn’t the numbering itself—it’s that the numbering lets a formal system represent facts about its own proofs. That bridge between syntax (symbols) and arithmetic (numbers) is the key insight that makes the incompleteness theorem work.

Godel also destroyed Hilbert's dream to finally set mathematics on a solid, invulnerable foundation for all time.

and this one from Numberphile:

The parts most people miss:

There are useful systems that ARE complete. That is, every proposition can be either proved or disproved within the system.

The boundary between decidable and undecidable systems is complicated with no particular pattern to it. Diophantine equations with five variables is undecidable.

Actually G proved that mathematics is either undecidable or contain contradictions. But no one really believes in the latter case.

Normally the undecidable propositions can be proved in some larger system. One way to do it is by adding an infinite number of axioms but that's gross so nobody does that.

BadgerBadger92 said:

The only part I understand is that math can’t prove everything.

The issue mainly concerns proving self-referential statements about the proving itself. This is usually of little concern for practical applications of math, but annoying to formalists who work on more abstract levels.