## Abstract

We demonstrate that the technology of Radin forcing can be used to transfer compactness properties at a weakly inaccessible but not strong limit cardinal to a strongly inaccessible cardinal. As an application, relative to the existence of large cardinals, we construct a model of set theory in which there is a strongly inaccessible cardinal (Formula presented.) that is (Formula presented.) - (Formula presented.) -stationary for all (Formula presented.) but not weakly compact. This is in sharp contrast to the situation in the constructible universe (Formula presented.), where (Formula presented.) being (Formula presented.) - (Formula presented.) -stationary is equivalent to (Formula presented.) being (Formula presented.) -indescribable. We also show that it is consistent that there is a cardinal (Formula presented.) such that (Formula presented.) is (Formula presented.) -stationary for all (Formula presented.) and (Formula presented.), answering a question of Sakai.

Original language | English |
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Article number | e12940 |

Journal | Journal of the London Mathematical Society |

Volume | 109 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2024 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2024 The Author(s). The Journal of the London Mathematical Society is copyright © London Mathematical Society.