Weakly compact cardinals and nonspecial Aronszajn trees

LEMMA 1. If A is a cardinal with cf A > w, then E\ implies that there is a A\+-Aronszajn tree with an w-ascent path, i.e. a sequence (xtc: a +) with each = (Xa: n +, Xn precedes 4 in the tree order for sufficiently large n. LEMMA 2. If A is a cardinal with cf A = w v+ is a monotone increasing function of a, then T is nonspecial. THEOREM 4. If A is uncountable, then 0Z implies that there is a nonspecial A+ -Aronszajn tree. THEOREM 5. If AK is an uncountable cardinal, ,c = A+, and r, is not (weakly compact) L, then there is a nonspecial rc-Aronszajn tree. Notation. CH is the continuum hypothesis. Souslin trees are S-trees; ic-Souslin trees are ic-S-trees. Aronszajn trees are A-trees; ic-Aronszajn trees are ic-A-trees. A ic-wide tree is a tree of height ic with node set a subset of ic, with no ic-branches. SH,K is the ic-Souslin Hypothesis: there are no ic-Souslin trees. If ic = A+ and T is a ic-wide-tree, then T is ic-special iff there is f: T -* A, such that for any chain C of T, f IC is one-to-one (a more general formulation of this allows us to generalize the notion to regular limit cardinals, but we have no use for the generalization here). SAH,I is the ic-special-Aronszajn Hypothesis: all ic-A-trees are ic-special. Thus, SAH,I = SH,. BA is Baumgartner's Axiom and BACH is the conjunction of BA and CH; see [8 and 9, or 10]. Other notation will be introduced as needed or is intended to be standard or else have a clear meaning. Introduction. In [7] we proved that CH + SHN2 is a large cardinal hypothesis: we obtained consistency strength at least that of the existence of an inaccessible (we will write inaccessible <,O.s CH + SHN2, and similar expressions; equiconsistency statements will use ) In [4], Laver and Shelah showed that CH + SH 2 <cons weakly compact; in fact their methods show that BACH + SAH 2 <cons weakly Received by the editors July 16, 1986 and, in revised form, March 10, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 03E05, 03E45, 03E55, 03E99, 04A20, 04A99. The first author was partially supported by a BSF grant. Both authors were partially supported