# 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)

Bib entry

@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} }