On ordinals accessible by infinitary languages

Saharon Shelah¹ ; Pauli Väisänen² ; Jouko Väänänen²

¹ Institute of Mathematics The Hebrew University Jerusalem, Israel and Department of Mathematics Rutgers University New Brunswick, NJ, U.S.A.
² Department of Mathematics University of Helsinki P.O. Box 4 FIN-00014, Finland

Fundamenta Mathematicae, Tome 186 (2005) no. 3, pp. 193-214

Résumé

Let $\lambda$ be an infinite cardinal number. The ordinal number $\delta(\lambda)$ is the least ordinal $\gamma$ such that if $\phi$ is any sentence of $L_{\lambda^+\omega}$, with a unary predicate $D$ and a binary predicate $\prec$, and $\phi$ has a model ${\cal M}$ with $\langle D^{\cal M},\prec^{\cal M}\rangle$ a well-ordering of type $\ge\gamma$, then $\phi$ has a model ${\cal M}'$ where $\langle D^{{\cal M}'},\prec^{{\cal M}'}\rangle$ is non-well-ordered. One of the interesting properties of this number is that the Hanf number of $L_{\lambda^+\omega}$ is exactly $\beth_{\delta(\lambda)}$. It was proved in \cite{BarwiseKunen1971} that if $\aleph_ 0 \lambda \kappa$ are regular cardinal numbers, then there is a forcing extension, preserving cofinalities, such that in the extension $2^ \lambda = \kappa$ and $\delta (\lambda) \lambda^{{+}{+}}$. We improve this result by proving the following: Suppose $\aleph_ 0 \lambda \theta \leq \kappa$ are cardinal numbers such that $\bullet$ $\lambda^{ \lambda} = \lambda$;$\bullet$ ${\rm cf}( \theta) \geq \lambda^+$ and $\mu^\lambda \theta$ whenever $\mu \theta$;$\bullet$ $\kappa^\lambda = \kappa$.Then there is a forcing extension preserving all cofinalities, adding no new sets of cardinality $ \lambda$, and such that in the extension $2^ \lambda = \kappa$ and $\delta( \lambda) = \theta$.

Détail

DOI : 10.4064/fm186-3-1

Keywords: lambda infinite cardinal number ordinal number delta lambda least ordinal gamma phi sentence lambda omega unary predicate binary predicate prec phi has model cal langle cal prec cal rangle well ordering type gamma phi has model cal where langle cal prec cal rangle non well ordered interesting properties number hanf number lambda omega exactly beth delta lambda proved cite barwisekunen aleph lambda kappa regular cardinal numbers there forcing extension preserving cofinalities extension lambda kappa delta lambda lambda improve result proving following suppose aleph lambda theta leq kappa cardinal numbers bullet lambda lambda lambda bullet theta geq lambda lambda theta whenever theta bullet kappa lambda kappa there forcing extension preserving cofinalities adding sets cardinality lambda extension lambda kappa delta lambda theta