Sh:614

• Džamonja, M., & Shelah, S. (2004). On the existence of universal models. Arch. Math. Logic, 43(7), 901–936.
• Abstract:
Suppose that \lambda=\lambda^{<\lambda}\ge\aleph_0, and we are considering a theory T. We give a criterion on T which is sufficient for the consistent existence of \lambda^{++} universal models of T of size \lambda^+ for models of T of size \le\lambda^+, and is meaningful when 2^{\lambda^+}>\lambda^{++}. In fact, we work more generally with abstract elementary classes. The criterion for the consistent existence of universals applies to various well known theories, such as triangle-free graphs and simple theories.

Having in mind possible aplpications in analysis, we further observe that for such \lambda, for any fixed \mu>\lambda^+ regular with \mu=\mu^{\lambda^+}, it is consistent that 2^\lambda=\mu and there is no normed vector space over {\mathbb Q} of size <\mu which is universal for normed vector spaces over {\mathbb Q} of dimension \lambda^+ under the notion of embedding h which specifies (a,b) such that \|h(x)\|/\|x\|\in (a,b) for all x.

• Version 2004-02-11_10 (51p) published version (36p)
Bib entry
@article{Sh:614,
author = {D{\v{z}}amonja, Mirna and Shelah, Saharon},
title = {{On the existence of universal models}},
journal = {Arch. Math. Logic},
fjournal = {Archive for Mathematical Logic},
volume = {43},
number = {7},
year = {2004},
pages = {901--936},
issn = {0933-5846},
mrnumber = {2096141},
mrclass = {03E35 (03C50 03C55 03C95)},
doi = {10.1007/s00153-004-0235-1},
note = {\href{https://arxiv.org/abs/math/9805149}{arXiv: math/9805149}},
arxiv_number = {math/9805149}
}