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 -