Sh:464
- Baldwin, J. T., Laskowski, M. C., & Shelah, S. (1993). Forcing isomorphism. J. Symbolic Logic, 58(4), 1291–1301. arXiv: math/9301208 DOI: 10.2307/2275144 MR: 1253923
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Abstract:
A forcing extension may create new isomorphisms between two models of a first order theory. Certain model theoretic constraints on the theory and other constraints on the forcing can prevent this pathology. A countable first order theory is classifiable if it is superstable and does not have either the dimensional order property or the omitting types order property. Shelah [Sh:c] showed that if a theory is classifiable then each model of cardinality is described by a sentence of . In fact this sentence can be chosen in the . ( is the result of enriching the language by adding for each a quantifier saying the dimension of a dependence structure is greater than .) The truth of such sentences will be preserved by any forcing that does not collapse cardinals and that adds no new countable subsets of . Hence, if two models of a classifiable theory of power are non-isomorphic, they are non-isomorphic after a -complete forcing. Here we show that the hypothesis of the forcing adding no new countable subsets of cannot be eliminated. In particular, we show that non-isomorphism of models of a classifiable theory need not be preserved by ccc forcings. - Version 1989-01-14_10 (18p) published version (12p)
Bib entry
@article{Sh:464, author = {Baldwin, John T. and Laskowski, Michael Chris and Shelah, Saharon}, title = {{Forcing isomorphism}}, journal = {J. Symbolic Logic}, fjournal = {The Journal of Symbolic Logic}, volume = {58}, number = {4}, year = {1993}, pages = {1291--1301}, issn = {0022-4812}, mrnumber = {1253923}, mrclass = {03C45 (03C55)}, doi = {10.2307/2275144}, note = {\href{https://arxiv.org/abs/math/9301208}{arXiv: math/9301208}}, arxiv_number = {math/9301208} }