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The strong soundness theorem for real closed fields and Hilbert’s Nullstellensatz in second order arithmetic

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Abstract.

By RCA 0 , we denote a subsystem of second order arithmetic based on Δ0 1 comprehension and Δ0 1 induction. We show within this system that the real number system R satisfies all the theorems (possibly with non-standard length) of the theory of real closed fields under an appropriate truth definition. This enables us to develop linear algebra and polynomial ring theory over real and complex numbers, so that we particularly obtain Hilbert’s Nullstellensatz in RCA 0 .

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Authors and Affiliations

  1. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Nobuyuki Sakamoto

  2. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Kazuyuki Tanaka

Authors
  1. Nobuyuki Sakamoto
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  2. Kazuyuki Tanaka
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Correspondence to Nobuyuki Sakamoto.

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Mathematics Subject Classification (2000): Primary 03F35; Secondary 03B30, 12D10

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Sakamoto, N., Tanaka, K. The strong soundness theorem for real closed fields and Hilbert’s Nullstellensatz in second order arithmetic. Arch. Math. Logic 43, 337–349 (2004). https://doi.org/10.1007/s00153-003-0206-y

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  • DOI: https://doi.org/10.1007/s00153-003-0206-y

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