Inconsistent class

A class $K$ of formulas in the language of a given formal theory having the property that there is a formula $\phi$ such that both $\phi$ and $\neg\phi$ (the negation of $\phi$) can be deduced from $K$ in this theory. In other words, if all formulas in $K$ are added to the axioms of the theory as new axioms, then in the theory obtained one can deduce both the formula $\phi$ and the formula $\neg\phi$.

References

[a1]	A. Grzegorczyk, "An outline of mathematical logic" , Reidel (1974)
[a2]	S.C. Kleene, "Introduction to metamathematics" , North-Holland & Noordhoff (1950) pp. Chapt. XIV