Strong theories, burden, and weight

We introduce the notion of the burden of a partial type in a complete first-order theory and call a theory strong if all types have almost finite burden. In a simple theory it is the supremum of the weights of all extensions of the type, and a simple theory is strong if and only if all types have finite weight. A theory without the independence property is strong if and only if it is strongly dependent. As a corollary, a stable theory is strongly dependent if and only if all types have finite weight. A strong theory does not have the tree property of the second kind.