my forty years on his shoulders - Department of Mathematics

40The standard formalization of “predicative mathematics” is due toFeferman/Schutte = FS. See (Feferman 1964,1968), (Feferman 1998). Poincare,Weyl, and others railed against impredicative mathematics. See (Weyl 1910),(Weyl 1987), (Feferman 1998 289-291), and (Foline 1992).THEOREM 8.2. (Friedman 2002b). Kruskal’s tree theorem cannot be proved inFS.KT goes considerably beyond FS, and an exact measure of KT is known.See (Rathjen, Weiermann 1993).J.B. Kruskal actually considered finite trees whose vertices are labeledfrom a wqo ≤*. The additional requirement on embeddings is that label(v) ≤*label(h(v)).THEOREM 8.3. (Kruskal 1960). The qo of finite trees as posets, with verticeslabeled from any given wqo, is a wqo.Labeled KT is considerably stronger, proof theoretically, than KT, evenwith only 2 labels, 0 ≤ 1. We have not seen a metamathematical analysis oflabeled KT.Note that KT is a Π 1 1 sentence and labeled KT is a Π 1 2 sentence.THEOREM 8.4. Labeled KT does not hold in the hyperarithmetic sets. In fact,RCA 0 + KT implies ATR 0 .A proof of Theorem 8.4 will appear in (Friedman, Montalban, Weiermannin preparation).