Abstract
Abstract In this paper we exposit some as yet unpublished results of Harvey Friedman. These results provide the most dramatic examples so far known of mathematically meaningful theorems of finite combinatorics which are unprovable in certain logical systems. The relevant logical systems, ATR 0 and Π 1 1 -CA 0 , are well known as relatively strong fragments of second order arithmetic. The unprovable combinatorial theorems are concerned with embeddability properties of finite trees. Friedman's methods are based in part on the existence of a close relationship between finite trees on the one hand, and systems of ordinal notations which occur in Gentzen-style proof theory on the other.
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