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

Nobuyuki Sakamoto; Kazuyuki Tanaka

doi:10.1007/s00153-003-0206-y

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

Nobuyuki Sakamoto, Kazuyuki Tanaka

SCI - Mathematics

Research output: Contribution to journal › Article

1 Citation (Scopus)

Abstract

By RCA₀, we denote a subsystem of second order arithmetic based on Δ₁⁰ comprehension and ∑₁ ⁰ 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 RCA₀.

Original language	English
Pages (from-to)	337-349
Number of pages	13
Journal	Archive for Mathematical Logic
Volume	43
Issue number	3
DOIs	https://doi.org/10.1007/s00153-003-0206-y
Publication status	Published - 2004 Apr 1

Keywords

Algebraically closed fields
Hilbert's Nullstellensatz
Real closed fields
Reverse mathematics
Second order arithmetic

ASJC Scopus subject areas

Philosophy
Logic

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10.1007/s00153-003-0206-y

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Sakamoto, N., & Tanaka, K. (2004). The strong soundness theorem for real closed fields and Hilbert's Nullstellensatz in second order arithmetic. Archive for Mathematical Logic, 43(3), 337-349. https://doi.org/10.1007/s00153-003-0206-y

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