View source for Inconsistent class

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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 $\neg\phi$.



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Grzegorczyk,   "An outline of mathematical logic" , Reidel  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.C. Kleene,   "Introduction to metamathematics" , North-Holland &amp; Noordhoff  (1950)  pp. Chapt. XIV</TD></TR></table>