可刻画基数的线性序与幂
Linear Orderings and Powers of Characterizable Cardinal
摘要：
本文回答了abs/1007.2426中的一个公开问题：如果M的Scott语句有基数Kappa的模型，但没有基数Kappa+的模型，则称可数模型M刻画无限基数Kappa.如果M按<线性排序，我们说线性排序（M，<）表征kappa。已知如果kappa是可表征的，则kappa+是可由线性排序表征的。此外，如果kappa可由具有大小为kappa的递增序列的稠密线性序来表征，则2^kappa是可表征的。我们表明，如果kappa是均匀的特征，那么kappa的特征是由一个密集的线性排序，而反过来失败。主要定理是：1）如果kappa> 2 ^lambda是可刻画基数，lambda是稠密线性序可刻画的，lambda是使得kappa ^lambda> kappa的最小基数，则kappa ^lambda也是可刻画的，2）如果aleph_alpha和kappa ^（aleph_alpha）是可刻画基数，则对于所有可数的beta，kappa ^（aleph_（alpha + beta））也是如此。结合这两个定理，我们得到：如果kappa> 2 ^（aleph_alpha）是可刻画基数，aleph_alpha是稠密线性序可刻画的，aleph_alpha是使得kappa ^（aleph_alpha）> kappa的最小基数，则对所有beta <alpha + omega_1，kappa ^（aleph_beta）是可刻画的。同样，如果kappa是一个可刻画基数，那么kappa ^（aleph_alpha）是可刻画的，对所有可数的alpha。
英文摘要：
The current paper answers an open question of abs/1007.2426 We say that a countable model M characterizes an infinite cardinal kappa, if the Scott sentence of M has a model in cardinality kappa, but no models in cardinality kappa plus. If M is linearly ordered by <, we will say that the linear ordering (M,<) characterizes kappa. It is known that if kappa is characterizable, then kappa plus is characterizable by a linear ordering. Also, if kappa is characterizable by a dense linear ordering with an increasing sequence of size kappa, then 2^kappa is characterizable. We show that if kappa is homogeneously characterizable, then kappa is characterizable by a dense linear ordering, while the converse fails. The main theorems are: 1) If kappa>2^lambda is a characterizable cardinal, lambda is characterizable by a dense linear ordering and lambda is the least cardinal such that kappa^lambda>kappa, then kappa^lambda is also characterizable, 2) if aleph_alpha and kappa^(aleph_alpha) are characterizable cardinals, then the same is true for kappa^(aleph_(alpha+beta)), for all countable beta. Combining these two theorems we get that if kappa>2^(aleph_alpha) is a characterizable cardinal, aleph_alpha is characterizable by a dense linear ordering and aleph_alpha is the least cardinal such that kappa^(aleph_alpha)>kappa, then for all beta<alpha+omega_1, kappa^(aleph_beta) is characterizable. Also if kappa is a characterizable cardinal, then kappa^(aleph_alpha) is characterizable, for all countable alpha.
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