On Hanf numbers of the infinitary order property

We study several cardinal, and ordinal--valued functions that are relatives of Hanf numbers. Let kappa be an infinite cardinal, and let T subseteq L_{kappa^+, omega} be a theory of cardinality <= kappa, and let gamma be an ordinal >= kappa^+. For example we look at (1) mu_{T}^*(gamma, kappa):= min {mu^* for all phi in L_{infinity, omega}, with rk(phi)< gamma, if T has the (phi, mu^*)-order property then there exists a formula phi'(x;y) in L_{kappa^+, omega}, such that for every chi >= kappa, T has the (phi', chi)-order property}; and (2) mu^*(gamma, kappa):= sup{mu_T^*(gamma, kappa)| T in L_{kappa^+,omega}}

A presentation theorem for continuous logic and Metric Abstract Elementary Classes

We give a presentation theorem for continuous first-order logic and Metric Abstract Elementary classes in terms of $L_{\omega_1, \omega}$ and Abstract Elementary Classes, respectively. This presentation is accomplished by analyzing dense subsets that are closed under functions. We extend this correspondence to types and saturation