On Hanf Numbers of the Infinitary Order Property Draft

We study several cardinal, and ordinal–valued functions that are relatives of Hanf numbers. Let κ be an infinite cardinality, and let T ⊆ Lκ+,ω be a theory of cardinality ≤ κ, and let γ be an ordinal ≥ κ. Consider (1) μT (γ, κ) := min{μ∗ : ∀φ ∈ L∞,ω , with rk(φ) < γ, if T has the (φ, μ∗)order property then there exists a formula φ ′ (x; y) ∈ Lκ+,ω , such that for every χ ≥ κ, T has the (φ , χ)-order property }. (2) μ∗(γ, κ) := sup{μT (γ, κ) | T ∈ Lκ+,ω}. We discuss several other related functions, sample results are: · It turns out that if T has the (φ, μ∗(γ, κ))-order propery for some φ ∈ L∞,ω , with rk(φ) < γ then for every χ > κ we have that I(χ, T ) = 2 holds. · For every κ and γ as above there exists an ordinal δ∗(γ, κ) such that μ∗(γ, κ) = iδ∗(γ,κ), · δ∗(γ, κ) ≤ (|γ|), · for κ with uncountable cofinality, we have that δ∗(γ, κ) > |γ| and · the ordinal δ∗(γ, κ) is bounded below by the Galvin–Hajnal rank of a reduced product. For many cardinalities we have better bounds, some of the bounds obtained using Shelah’s pcf theory. The function μ∗(γ, κ) is used to compute bounds to the values of the function μ(λ, κ) we studied in a previous paper. Date: June 9, 1998. The authors thank the United States Israel Binational Science foundation for supporting this research as well as the Mathematics department of Rutgers University where part of this research was carried out. Partially supported by the United States-Israel Binational Science Foundation and by the National Science Fundation, NSF-DMS97-04477. This paper replaces item # 259 from Shelah’s list of publications.