## Abstract

This paper studies structural consequences of supercompactness of ω_{1} under ZF. We show that the Axiom of Dependent Choice (DC) follows from “ ω_{1} is supercompact”. “ ω_{1} is supercompact” also implies that AD^{+}, a strengthening of the Axiom of Determinacy (AD), is equivalent to AD_{R}. It is shown that “ ω_{1} is supercompact” does not imply AD. The most one can hope for is Suslin determinacy. We show that this follows from “ ω_{1} is supercompact” and Hod Pair Capturing (HPC), an inner-model theoretic hypothesis that imposes certain smallness conditions on the universe of sets. “ ω_{1} is supercompact” on its own implies that every Suslin set is the projection of a determined (in fact, homogenously Suslin) set. “ ω_{1} is supercompact” also implies all sets in the Chang model have all the usual regularity properties, like Lebesgue measurability and the Baire property.

Original language | English |
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Title of host publication | Advances in Mathematical Logic - Dedicated to the Memory of Professor Gaisi Takeuti, SAML 2018, Selected, Revised Contributions |

Editors | Toshiyasu Arai, Makoto Kikuchi, Satoru Kuroda, Mitsuhiro Okada, Teruyuki Yorioka |

Publisher | Springer |

Pages | 27-45 |

Number of pages | 19 |

ISBN (Print) | 9789811641725 |

DOIs | |

Publication status | Published - 2021 |

Event | Symposium on Advances in Mathematical Logic, SAML 2018 - Kobe, Japan Duration: 2018 Sep 18 → 2018 Sep 20 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 369 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Conference

Conference | Symposium on Advances in Mathematical Logic, SAML 2018 |
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Country/Territory | Japan |

City | Kobe |

Period | 18/9/18 → 18/9/20 |

## Keywords

- Axiom of determinacy
- Large cardinal properties
- Supercompactness
- ω

## ASJC Scopus subject areas

- Mathematics(all)

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