紧致性定理在群论中的应用

Applications of the Compactness Theorem in Group Theory

文章用模型论中的紧致性定理证明了若L中理论T有任意可数阶的Abel群模型,则T有无扭Abel群模型;若一个语句φ在任意一个无扭Abel群中真,则对任意大的自然数n,存在自然数m>n,使φ在m阶Abel群中真.最后证明了无扭Abel群不能有限公理化.

The author proved by the compactness theorem that if the theory of T with arbitrary countable order Abel group model in a formal language L then T has a torsion free group Abel model .If aφstatement is true in an arbitrary torsion free group Abel , then for a natural number n with arbitrarily large , there is a natural number m , which is larger than the natural number n , such that theφstatement is true in a group Abel with order m .At last , the author proved that a torsion free group Abel is not finitely axiomatizable in a formal language L .