On the existence of universal models
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 trianglefree graphs and simple theories. Having in mind possible applications 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 ${\Bbf Q}$ of size $<\mu$ which is universal for normed vector spaces over ${\Bbf Q}$ of dimension $\lambda^+$ under the notion of embedding $h$ which specifies $(a,b)$ such that $\norm{h(x)}/\norm{x}\in (a,b)$ for all $x$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 1998
 DOI:
 10.48550/arXiv.math/9805149
 arXiv:
 arXiv:math/9805149
 Bibcode:
 1998math......5149D
 Keywords:

 Mathematics  Logic;
 Mathematics  Functional Analysis
 EPrint:
 Archive for Mathematical Logic, vol. 43 (2004), pg. 901936