Mathematics – Logic
Scientific paper
2010-12-15
Mathematics
Logic
Version 01: 20 pages, no figures, submitted for publication in November 2010
Scientific paper
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
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