Across many fields of math, and related fields like logic and computer science, there are incompleteness theorems that state a system cannot be both consistent and complete. Some examples include (simplifying a lot):
- Gödel incompleteness theorem: "The i-th statement cannot be proved", where the i-th statement is that text.
- Logical statement: "This sentence is false" cannot be categorized as either true or false.
- Linguistics: "Autological" and "Heterological" cannot be categorized as either autological or heterological.
- Linguistics: Is this true: "is not true of itself" is not true of itself.
- Set theory: A set of all sets that do not contain themselves. Does this set contain itself?
- Computer science: The halting problem defines a hypothetical device that can determine if a program will stop. If you insert the device into itself such that it never halts on one condition, it cannot be decided.
I have noticed a pattern, all examples of incompleteness I have seen involve circular references. Some of them involve indirect references, such as the Gödel incompleteness theorem, but they still reference themselves.
Is this a requirement of incompleteness theorems in general, that the only way it can be proven that consistent formal systems are incomplete is if that system is allowed to reference its own definitions? Are logical statements complete if the statements are not allowed to refer to themselves? Is set theory complete if we only look at sets that do not have conditions defined based on sets containing sets?
I have also found this question about the Gödel incompleteness theorem specifically, but it does not quite answer this very general question. The top answer seems to be explaining how it arrives at a self-reference indirectly.
