TY - GEN

T1 - On Supercompactness of ω1

AU - Ikegami, Daisuke

AU - Trang, Nam

N1 - Funding Information:
Acknowledgements We would like to thank Hugh Woodin for communicating his insights on this subject as well as his results concerning the model C+. We are grateful to the anonymous referee for numerous helpful comments. The first author would like to thank the Japan Society for the Promotion of Science (JSPS) for its generous support through the grant with JSPS KAKENHI Grant Number 15K17586 and 19K03604. He is also grateful to the Sumitomo Foundation for its generous support through Grant for Basic Science Research. The second author would like to thank the National Science Foundation (NSF) for its generous support through Grants DMS-1565808 and DMS-1849295.
Publisher Copyright:
© 2021, Springer Nature Singapore Pte Ltd.

PY - 2021

Y1 - 2021

N2 - This paper studies structural consequences of supercompactness of ω1 under ZF. We show that the Axiom of Dependent Choice (DC) follows from “ ω1 is supercompact”. “ ω1 is supercompact” also implies that AD+, a strengthening of the Axiom of Determinacy (AD), is equivalent to ADR. It is shown that “ ω1 is supercompact” does not imply AD. The most one can hope for is Suslin determinacy. We show that this follows from “ ω1 is supercompact” and Hod Pair Capturing (HPC), an inner-model theoretic hypothesis that imposes certain smallness conditions on the universe of sets. “ ω1 is supercompact” on its own implies that every Suslin set is the projection of a determined (in fact, homogenously Suslin) set. “ ω1 is supercompact” also implies all sets in the Chang model have all the usual regularity properties, like Lebesgue measurability and the Baire property.

AB - This paper studies structural consequences of supercompactness of ω1 under ZF. We show that the Axiom of Dependent Choice (DC) follows from “ ω1 is supercompact”. “ ω1 is supercompact” also implies that AD+, a strengthening of the Axiom of Determinacy (AD), is equivalent to ADR. It is shown that “ ω1 is supercompact” does not imply AD. The most one can hope for is Suslin determinacy. We show that this follows from “ ω1 is supercompact” and Hod Pair Capturing (HPC), an inner-model theoretic hypothesis that imposes certain smallness conditions on the universe of sets. “ ω1 is supercompact” on its own implies that every Suslin set is the projection of a determined (in fact, homogenously Suslin) set. “ ω1 is supercompact” also implies all sets in the Chang model have all the usual regularity properties, like Lebesgue measurability and the Baire property.

KW - Axiom of determinacy

KW - Large cardinal properties

KW - Supercompactness

KW - ω

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U2 - 10.1007/978-981-16-4173-2_2

DO - 10.1007/978-981-16-4173-2_2

M3 - Conference contribution

AN - SCOPUS:85124669380

SN - 9789811641725

T3 - Springer Proceedings in Mathematics and Statistics

SP - 27

EP - 45

BT - Advances in Mathematical Logic - Dedicated to the Memory of Professor Gaisi Takeuti, SAML 2018, Selected, Revised Contributions

A2 - Arai, Toshiyasu

A2 - Kikuchi, Makoto

A2 - Kuroda, Satoru

A2 - Okada, Mitsuhiro

A2 - Yorioka, Teruyuki

PB - Springer

T2 - Symposium on Advances in Mathematical Logic, SAML 2018

Y2 - 18 September 2018 through 20 September 2018

ER -