Sh:66

• Shelah, S. (1978). End extensions and numbers of countable models. J. Symbolic Logic, 43(3), 550–562. DOI: 10.2307/2273531 MR: 503792
• Abstract:
  The answer to the question from page 562 (the end). is negative; have known a solution but not sure if have Not record it.

  For any countable model M with countable vocabulary with predicates only. Not including < and E. First we choose a function F from Q the rationals onto M such that the pre-image of any element is dense Second we define a model N Universe. The rationals <. Is interpreted. As the rational order E is interpreted as the equivalence relation xEy iff F(x)=F(y) For any predicate P of the vocabulary of M is interpreted as it’s pre-image by F No Th(N) is a countable fo theory with the same number of countable models up to isomorphism as Th(M) So we are done giving a negative answer to the question

• published version (14p)

@article{Sh:66,
 author = {Shelah, Saharon},
 title = {{End extensions and numbers of countable models}},
 journal = {J. Symbolic Logic},
 fjournal = {The Journal of Symbolic Logic},
 volume = {43},
 number = {3},
 year = {1978},
 pages = {550--562},
 issn = {0022-4812},
 mrnumber = {503792},
 mrclass = {03C15},
 doi = {10.2307/2273531}
}