# Sh:66

• Shelah, S. (1978). End extensions and numbers of countable models. J. Symbolic Logic, 43(3), 550–562.
• 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}
}