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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