Well Partial Orders

This article provides a (personal) survey on well partial orders. We mainly concentrate on aspects which we find most interesting and apologize for omitting many other important results. Starting with recalling basic facts from De Jongh and Parikh’s fundamental paper on maximal order types we survey some related key results by Diana Schmidt. Then we discuss generalized trees, their embeddability relation and their associated maximal order types. These results can be condensed neatly into the formula \(o(T(W))=\vartheta (o(W(\Omega )))\) . At the end we cover Friedman style miniaturizations of Kruskal’s theorem and their associated phase transitions.