关于整数环的无限n素元组归纳扩环的多样性

用模型论方法证明了: 对于整数环I及每个正整数n及每一可能类型的n素元组而言,存在不可数无限多个其1阶性质互不全同的含有无限多该类型的n素元组的归纳扩环(文中简称该类型的nT-环),并且存在无限多个正整数对,对每个这样的正整数对(a,b),存在I的nT-扩环R,它适合加a的归纳法,而不适合加b的归纳法. 还证明了存在该类型的nT-环R,R的每个元素是3平方和和4立方和.

ON THE ABUNDANCE OF THE INDUCTIVE EXTENSIONS OF THE RING OF INTEGERS WITH INFINITELY MANY n-PRIME TUPLES

It is shown by model-theoretic methods that the following results are valid for any positive integer n and any possible type of n-prime tuples: 1 There exist uncountably infinitely many inductive extensions of the ring I of integers with infinitely many such n-prime tuples(called nT-rings in the following)and these rings are not equivalent to each other in first order logic. 2 There exist infinitely many couples of positive integers such that for each couple(a, b), there exists an nT-ring R which satisfies the inductive principle with step a and does not satisfy the inductive principle with step b. 3 There exist nT-rings R such that every element of R is a sum of 3 squares and a sum of 4 cubes.