It is very hard to detect self-referential statements and restricting yourself to "non-self-referential statements" might be quite severe.
Given a set of "domino" tiles - each having a top and a bottom. Each top and bottom has some word on it - these words can only use the letters "a" and "b".
You can duplicate domino tiles and also align tiles so that all tops and all bottoms are aligned.
Now, given a finite set of such tiles, can you say whether there is an alignment so that all tops concatenated, read from right to left, equal all bottoms concatenated?
In fact, given such a set of tiles S, you can easily create a formula P(S) that is true iff such a valid alignment does not exist. Obviously this formula is true for some sets of tiles and false for others.
Now the funny thing: Given a (correct) fixed theory T in which you can state P(S) for every S and in which proofs can be computationally checked, there must be infinitely many S so that P(S) is true, but cannot be proved in T. Thus such theory T is incomplete.
Where is the self-reference?
This problem is also known as the Post correspondence problem (PCP). The halting problem can be reduced to it, which is not decidable. If T was complete, you could enumerating all proofs and see whether they correctly proof P(S) or its negation. Due to its completeness you would eventually find a proof for either of them and thus you could decide the halting problem.
Given a set of "domino" tiles - each having a top and a bottom. Each top and bottom has some word on it - these words can only use the letters "a" and "b". You can duplicate domino tiles and also align tiles so that all tops and all bottoms are aligned.
Now, given a finite set of such tiles, can you say whether there is an alignment so that all tops concatenated, read from right to left, equal all bottoms concatenated?
In fact, given such a set of tiles S, you can easily create a formula P(S) that is true iff such a valid alignment does not exist. Obviously this formula is true for some sets of tiles and false for others.
Now the funny thing: Given a (correct) fixed theory T in which you can state P(S) for every S and in which proofs can be computationally checked, there must be infinitely many S so that P(S) is true, but cannot be proved in T. Thus such theory T is incomplete.
Where is the self-reference?
This problem is also known as the Post correspondence problem (PCP). The halting problem can be reduced to it, which is not decidable. If T was complete, you could enumerating all proofs and see whether they correctly proof P(S) or its negation. Due to its completeness you would eventually find a proof for either of them and thus you could decide the halting problem.