## 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 $${\mathcal P}$$ denote the power-set operation and T the Kripke-Platek axioms in the language $$\{\epsilon$$,$${\mathcal P}\}$$ 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 $$\phi \in \Delta_ 0({\mathcal P})$$, then $$V\vDash (\exists x\in R_{\lambda})\forall y\phi.$$
(c) The least cardinal $$\mu$$ such that whenever $$\psi \in \Sigma_ 2({\mathcal P})$$ and $$V\vDash \psi$$, then $$R_{\mu}\vDash \psi$$; and whenever $$\theta (x)\in \Sigma_ 1({\mathcal P})$$, $$a\in R_{\mu}$$, and $$V\vDash \theta (a)$$, then $$R_{\mu}\vDash \theta (a).$$
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 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:

 03C85 Second- and higher-order model theory 03C95 Abstract model theory

### Keywords:

Hanf number; second order logic; Kripke-Platek axioms

### Citations:

Zbl 0281.02020; Zbl 0293.02039
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