TY - GEN
T1 - Choiceless Löwenheim–Skolem Property and Uniform Definability of Grounds
AU - Usuba, Toshimichi
N1 - Funding Information:
Acknowledgements The author would like to thank Asaf Karagila for his many valuable comments. The author also thank the referee who gives the author many corrections, and Daisuke Ikegami who pointed out the failure of SVC in Chang’s model. This research was supported by JSPS KAKENHI Grant Nos. 18K03403 and 18K03404.
Publisher Copyright:
© 2021, Springer Nature Singapore Pte Ltd.
PY - 2021
Y1 - 2021
N2 - In this paper, without the axiom of choice, we show that if a certain downward Löwenheim–Skolem property holds then all grounds are uniformly definable. We also prove that the axiom of choice is forceable if and only if the universe is a small extension of some transitive model of ZFC.
AB - In this paper, without the axiom of choice, we show that if a certain downward Löwenheim–Skolem property holds then all grounds are uniformly definable. We also prove that the axiom of choice is forceable if and only if the universe is a small extension of some transitive model of ZFC.
KW - Axiom of choice
KW - Forcing method
KW - Set-theoretic geology
UR - http://www.scopus.com/inward/record.url?scp=85124674323&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85124674323&partnerID=8YFLogxK
U2 - 10.1007/978-981-16-4173-2_8
DO - 10.1007/978-981-16-4173-2_8
M3 - Conference contribution
AN - SCOPUS:85124674323
SN - 9789811641725
T3 - Springer Proceedings in Mathematics and Statistics
SP - 161
EP - 179
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 -