# Sh:648

• Shelah, S., & Villaveces, A. (2021). The Hart-Shelah example, in stronger logics. Ann. Pure Appl. Logic, 172(6), 102958, 23.
• Abstract:
We generalize the Hart-Shelah example from [HaSh:323] to higher infinitary logics. We build, for each natural number k\geq 2 and for each infinite cardinal \lambda, a sentence \psi_k^\lambda of the logic L_{(2^\lambda)^+,\omega} that (modulo mild set theoretical hypotheses around \lambda and assuming 2^\lambda < \lambda^{+k}) is categorical in \lambda^+,\dots,\lambda^{+k-1} but not in \beth_{k+1}(\lambda)^+ (or beyond); we study the dimensional encoding of combinatorics involved in the construction of this sentence and study various model-theoretic properties of the resulting abstract elementary class {\mathcal K}^*(\lambda,k)=(Mod(\psi_k^\lambda),\prec_{(2^\lambda)^+,\omega}) in the finite interval of cardinals \lambda,\lambda^+,\dots,\lambda^{+k}.
• Version 2021-01-30 (31p)
Bib entry
@article{Sh:648,
author = {Shelah, Saharon and Villaveces, Andr{\'e}s},
title = {{The {H}art-{S}helah example, in stronger logics}},
journal = {Ann. Pure Appl. Logic},
fjournal = {Annals of Pure and Applied Logic},
volume = {172},
number = {6},
year = {2021},
pages = {102958, 23},
issn = {0168-0072},
mrnumber = {4216281},
mrclass = {03C48 (03C35 03C55 03C75 05C69)},
doi = {10.1016/j.apal.2021.102958},
note = {\href{https://arxiv.org/abs/math/0404258}{arXiv: math/0404258}},
arxiv_number = {math/0404258}
}