 The third condition  stipulates that \(Cn(X)\) is equal to the union of the set of formulas derivable from finite subsets of \(X\). Tarski took these properties to define the notion of consequence operation axiomatically. In fact, he added that there is a formula \(x\) such that \(Cn(\{x\})\) is the set \(A\) of all the formulas and that this set must be finite or of the cardinality of the set of the natural numbers. Condition (3) implies the weaker, and important, condition of monotonicity