The possibility of circle references in human language is totally independent of consistancy of mathematics. It's common sense and it can also be proved.
"This statement about axiomatic set theory is false."
"The class of all classes that do not include themselves, do not include itself."
"Next sentence is false. Previous sentence is true."
Statements with circle references may conflict with the Aristotelian logic of terms, but do not interfere with mathematical deduction, because there is an logically equivalent and consistent three valued logic, which admit circle references.
The trick is to split 'false' into two logically equivalent but different alternatives: 'false' and 'absurd' in a systematic algebraic way, which results in a unique commutative semiring which can be interpreted as a three valued logic that preserve all tautologies and rules of inference. All theorems can be proved and absurd statements exists besides false statements and as harmless.