system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: The Incompleteness Theorem.
gpt
The first of Godel's incompleteness theorems has to do with what happens when we pair a comprehensive theory of mathematics (we'll call it T) with a corresponding statement called its "Godel Sentence" (we'll call it G). The Godel Sentence for any given mathematical model is as follows:

"This statement is not provable by the axioms of the referenced model." 

Using symbolic notation:

G asserts: "G cannot be proved within the theory T."

Here, the Godel Sentence acts as a "true but not provable within T" statement. The interaction between a theory and its Godel Sentence is as follows:

If you could, by the rules of mathematical logic and inference, extract the "truth" of the statement G by using the axioms of the Theory T, then the theory has just contradicted itself. Think about it: if T proves something that says "T cannot prove me" to be true, then T must be inconsistent, because it has produced a contradiction. 

On the other hand, say that G actually holds up, and there is no possible way to logically infer the statement G with the axioms of theory T. Now, T is still consistent, because it has avoided the contradiction of accepting G under its paradigm, but by doing so, it has demonstrated the truth of G, because, like G said, "I cannot be proven in T". Hence, the theory T is incomplete, because there exists a true statement that it cannot possibly prove. 

These two scenarios present the fact that for any set of axioms that make up a theory T, the theory must either be inconsistent (and therefore useless in terms of mathematical logic), or it must be incomplete because of its inability to prove some demonstrably true statement. When this happens, our first instinct would be to just add G to our list of axioms of T, but by doing so, we effectively generate a different and distinct theory T2, which has its own different Godel sentence G2. This continues on and on, and thus any axiomatic formal theory of mathematics using our currently accepted rules of logic and inference will always be "incomplete".