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Choiceless Löwenheim–Skolem Property and Uniform Definability of Grounds

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Advances in Mathematical Logic (SAML 2018)

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Abstract

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 \(\mathsf {ZFC}\).

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Notes

  1. 1.

    In [4], our M(X) is referred to M[X].

  2. 2.

    Actually Kripke-Platek set-theory is sufficient.

  3. 3.

    Of course this definition cannot be formalized within \(\mathsf {ZF}\), so we will use it informally.

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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 \(\mathrm {SVC}\) in Chang’s model. This research was supported by JSPS KAKENHI Grant Nos. 18K03403 and 18K03404.

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Authors and Affiliations

  1. Faculty of Fundamental Science and Engineering, Waseda University, Okubo 3-4-1, Shinjyuku, Tokyo, 169-8555, Japan

    Toshimichi Usuba

Authors
  1. Toshimichi Usuba
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Correspondence to Toshimichi Usuba .

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Editors and Affiliations

  1. The University of Tokyo, Tokyo, Japan

    Toshiyasu Arai

  2. Kobe University, Kobe, Japan

    Makoto Kikuchi

  3. Gunma Prefectural Women’s University, Gunma, Japan

    Satoru Kuroda

  4. Keio University, Tokyo, Japan

    Mitsuhiro Okada

  5. Shizuoka University, Shizuoka, Japan

    Teruyuki Yorioka

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Usuba, T. (2021). Choiceless Löwenheim–Skolem Property and Uniform Definability of Grounds. In: Arai, T., Kikuchi, M., Kuroda, S., Okada, M., Yorioka, T. (eds) Advances in Mathematical Logic. SAML 2018. Springer Proceedings in Mathematics & Statistics, vol 369. Springer, Singapore. https://doi.org/10.1007/978-981-16-4173-2_8

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