1.  27
    Uniform Versions of Some Axioms of Second Order Arithmetic.Nobuyuki Sakamoto & Takeshi Yamazaki - 2004 - Mathematical Logic Quarterly 50 (6):587-593.
    In this paper, we discuss uniform versions of some axioms of second order arithmetic in the context of higher order arithmetic. We prove that uniform versions of weak weak König's lemma WWKL and Σ01 separation are equivalent to over a suitable base theory of higher order arithmetic, where is the assertion that there exists Φ2 such that Φf1 = 0 if and only if ∃x0 for all f. We also prove that uniform versions of some well-known theorems are equivalent to (...)
    Direct download (4 more)  
    Export citation  
    Bookmark   7 citations  
  2.  15
    The Jordan Curve Theorem and the Schönflies Theorem in Weak Second-Order Arithmetic.Nobuyuki Sakamoto & Keita Yokoyama - 2007 - Archive for Mathematical Logic 46 (5-6):465-480.
    In this paper, we show within ${\mathsf{RCA}_0}$ that both the Jordan curve theorem and the Schönflies theorem are equivalent to weak König’s lemma. Within ${\mathsf {WKL}_0}$ , we prove the Jordan curve theorem using an argument of non-standard analysis based on the fact that every countable non-standard model of ${\mathsf {WKL}_0}$ has a proper initial part that is isomorphic to itself (Tanaka in Math Logic Q 43:396–400, 1997).
    Direct download (3 more)  
    Export citation  
    Bookmark   3 citations  
  3.  16
    The Strong Soundness Theorem for Real Closed Fields and Hilbert?S Nullstellensatz in Second Order Arithmetic.Nobuyuki Sakamoto & Kazuyuki Tanaka - 2004 - Archive for Mathematical Logic 43 (3):337-349.
    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.
    Direct download (3 more)  
    Export citation  
    Bookmark   1 citation