 A set of propositions \(\mathbf{F}\) is a  field (sometimes  algebra) iff \(\mathbf{F}\) contains \(W\) and it is closed under intersection, union and complementation. That is to say that if \(A,B\) are both elements of \(\mathbf{F}\) then \(W,A\cup B,A\cap B\) and \(\neg A\) are also elements of \(\mathbf{F}.\) A set of propositions \(\mathbf{F}\) is a  \(\sigma\)-field (sometimes  \(\sigma\)-algebra) iff it is a field that is closed under  countable intersections, i.e. if \(\mathbf{S}\subseteq \mathbf{F}\) is a countable collection of propositions, then the intersection of all its elements \(\cap \mathbf{S}\) is also an element of \(\mathbf{F}.\) That definition implies that a \(\sigma\)-field is also closed under countable unions. It is not difficult to prove that the intersection of \(\sigma\)-fields is also a \(\sigma\)-field. That implies that every collection of propositions \(\mathbf{F}\) generates \(\sigma(\mathbf{F})\), the least \(\sigma\)-field containing \(\mathbf{F}\), by intersecting the set of all \(\sigma\)-fields containing \(\mathbf{F}\).