# Sh:323

- Hart, B. T., & Shelah, S. (1990).
*Categoricity over P for first order T or categoricity for \phi\in\mathcal L_{\omega_1\omega} can stop at \aleph_k while holding for \aleph_0,\cdots,\aleph_{k-1}*. Israel J. Math.,**70**(2), 219–235. arXiv: math/9201240 DOI: 10.1007/BF02807869 MR: 1070267 -
Abstract:

Suppose L is a relational language and P\in L is a unary predicate. If M is an L-structure then P(M) is the L-structure formed as the substructure of M with domain \{a: M\models P(a)\}. Now suppose T is a complete first order theory in L with infinite models. Following Hodges, we say that T is relatively \lambda-categorical if whenever M, N\models T, P(M)=P(N), |P(M)|=\lambda then there is an isomorphism i:M\rightarrow N which is the identity on P(M). T is relatively categorical if it is relatively \lambda-categorical for every \lambda. The question arises whether the relative \lambda-categoricity of T for some \lambda>|T| implies that T is relatively categorical.In this paper, we provide an example, for every k>0, of a theory T_k and an L_{\omega_1\omega} sentence \varphi_k so that T_k is relatively \aleph_n-categorical for n < k and \varphi_k is \aleph_n-categorical for n<k but T_k is not relatively \beth_k-categorical and \varphi_k is not \beth_k-categorical.

- published version (17p)

Bib entry

@article{Sh:323, author = {Hart, Bradd T. and Shelah, Saharon}, title = {{Categoricity over $P$ for first order $T$ or categoricity for $\phi\in\mathcal L_{\omega_1\omega}$ can stop at $\aleph_k$ while holding for $\aleph_0,\cdots,\aleph_{k-1}$}}, journal = {Israel J. Math.}, fjournal = {Israel Journal of Mathematics}, volume = {70}, number = {2}, year = {1990}, pages = {219--235}, issn = {0021-2172}, doi = {10.1007/BF02807869}, mrclass = {03C35 (03C75)}, mrnumber = {1070267}, mrreviewer = {Wiktor Da\'nko}, doi = {10.1007/BF02807869}, note = {\href{https://arxiv.org/abs/math/9201240}{arXiv: math/9201240}}, arxiv_number = {math/9201240} }