K-Suslin树同ZF+┐AC的协调性问题

THE PROBLEM OF K-SUSLIN TREE BEING CONSISTENT WITH ZF+┐AC

本文研究K—Suslin树(K>ω_1)同ZF+┐AC的相对协调性问题。证明了这样的结论:“存在K—Suslin树+ZF+┐AC”(K>ω_1且为后继基数)相对于ZF系统是协调的。证明方法是在ZFC的一个可数传递模型M(M中存在K—Suslin树T)和其generic扩张M[G]之间构造一个模型N,即M(?)N(?)M[G],使得在N中T仍为K—Suslin树且N(?)+ZF+┐AC,于是便有N(?)存在K—Suslin树+ZF+┐AC。该问题的证明过程中主要采用了力迫法。

This paper ciscusses the problem that K-Suslin tree is consistent with ZF+ (?) AC andobtains this result: there is a K-Suslin tree+ ZF+ (?) AC is consistent relative to ZF, where K>ω_1 and is asuccessor cardinal. The author constructs a model N between a ountable transitive model M and the gener-ic extension M [G] such that there is a K-Suslin tree in N and N (?) ZF+ (?) AC. Hence. N (?) (?) there is aK-Suslin tree+ ZF+ (?) AC. Foreing, the principal method, plays an importhat role in proving this problem.