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A note on the Hanf number of second-order logic. (English) Zbl 0644.03021

The Hanf number of second order logic is the least cardinal κ such that every sentence of second order logic that has a model of power at least κ has arbitrarily large models. Let 𝒫 denote the power-set operation and T the Kripke-Platek axioms in the language {ϵ,𝒫} augmented with the power-set axiom and the axiom that every well-ordering is isomorphic to an ordinal. Consider the following cardinals:

(a) The Hanf-number κ of second order logic,

(b) The least cardinal λ such that whenever Vxyϕ with ϕΔ 0 (𝒫), then V(xR λ )yϕ·

(c) The least cardinal μ such that whenever ψΣ 2 (𝒫) and Vψ, then R μ ψ; and whenever θ(x)Σ 1 (𝒫), aR μ , and Vθ(a), then R μ θ(a)·

It is proved in T that if any of the cardinals (a)-(c) exists, they all exist and are equal. The result complements and builds on related results by J. Barwise [J. Symb. Logic 37, 588-594 (1972; Zbl 0281.02020)] and H. Friedman [ibid. 39, 318-324 (1974; Zbl 0293.02039)].

Reviewer: J.Väänänen
MSC:
03C85Second- and higher-order model theory
03C95Abstract model theory