### On Hanf numbers of the infinitary order property

by Grossberg and Shelah. [GrSh:259]

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_{infty, 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}} .

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