A very nice generalization of indexing by a well-ordered set and using induction to prove something is provided by the notion of a $\kappa$-good tree for $\kappa$ a (usually infinite) regular cardinal [see this MO question](http://mathoverflow.net/questions/28871/k-good-trees-and-k-compactness-of-colimits-over-k-small-downwards-closed-subposet).  To see it put to use in homotopy theory, take a look at Lurie's _Higher topos theory_ chapter 1.5.