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


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