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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 $\kappa$ such that every sentence of second order logic that has a model of power at least $\kappa$ has arbitrarily large models. Let ${\cal P}$ denote the power-set operation and T the Kripke-Platek axioms in the language $\{\epsilon$,${\cal P}\}$ 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 $\kappa$ of second order logic, (b) The least cardinal $\lambda$ such that whenever $V\vDash \exists x\forall y\phi$ with $\phi \in \Delta\sb 0({\cal P})$, then $V\vDash (\exists x\in R\sb{\lambda})\forall y\phi.$ (c) The least cardinal $\mu$ such that whenever $\psi \in \Sigma\sb 2({\cal P})$ and $V\vDash \psi$, then $R\sb{\mu}\vDash \psi$; and whenever $\theta (x)\in \Sigma\sb 1({\cal P})$, $a\in R\sb{\mu}$, and $V\vDash \theta (a)$, then $R\sb{\mu}\vDash \theta (a).$ 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 {\it J. Barwise} [J. Symb. Logic 37, 588-594 (1972; Zbl 0281.02020)] and {\it H. Friedman} [ibid. 39, 318-324 (1974; Zbl 0293.02039)].
Reviewer: J.Väänänen

03C85Second- and higher-order model theory
03C95Abstract model theory
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