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Forcing the Mapping Reflection Principle by finite approximations

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Abstract

Moore introduced the Mapping Reflection Principle and proved that the Bounded Proper Forcing Axiom implies that the size of the continuum is \(\aleph _2\). The Mapping Reflection Principle follows from the Proper Forcing Axiom. To show this, Moore utilized forcing notions whose conditions are countable objects. Chodounský–Zapletal introduced the Y-Proper Forcing Axiom that is a weak fragments of the Proper Forcing Axiom but implies some important conclusions from the Proper Forcing Axiom, for example, the P-ideal Dichotomy. In this article, it is proved that the Y-Proper Forcing Axiom implies the Mapping Reflection Principle by introducing forcing notions whose conditions are finite objects.

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Acknowledgements

The authors would like to thank the referee for giving us many useful comments.

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Authors and Affiliations

  1. Mathematics, Nanzan University, 18 Yamazato-cho, Showa-ku, Nagoya, 466-8673, Japan

    Tadatoshi Miyamoto

  2. Faculty of Science, Shizuoka University, Ohya 836, Shizuoka, 422-8529, Japan

    Teruyuki Yorioka

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  1. Tadatoshi Miyamoto
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  2. Teruyuki Yorioka
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Correspondence to Teruyuki Yorioka.

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Teruyuki Yorioka is supported by JSPS KAKENHI Grant Number JP18K03393.

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Miyamoto, T., Yorioka, T. Forcing the Mapping Reflection Principle by finite approximations. Arch. Math. Logic 60, 737–748 (2021). https://doi.org/10.1007/s00153-020-00756-1

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