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  1. Measure theory and weak König's lemma.Xiaokang Yu & Stephen G. Simpson - 1990 - Archive for Mathematical Logic 30 (3):171-180.
    We develop measure theory in the context of subsystems of second order arithmetic with restricted induction. We introduce a combinatorial principleWWKL (weak-weak König's lemma) and prove that it is strictly weaker thanWKL (weak König's lemma). We show thatWWKL is equivalent to a formal version of the statement that Lebesgue measure is countably additive on open sets. We also show thatWWKL is equivalent to a formal version of the statement that any Borel measure on a compact metric space is countably additive (...)
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  • Fixed point theory in weak second-order arithmetic.Naoki Shioji & Kazuyuki Tanaka - 1990 - Annals of Pure and Applied Logic 47 (2):167-188.
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  • 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.
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  • 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).
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  • The mean value theorem in second order arithmetic.Christopher S. Hardin & Daniel J. Velleman - 2001 - Journal of Symbolic Logic 66 (3):1353-1358.
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  • Subsystems of Second Order Arithmetic.Stephen George Simpson - 1999 - Springer Verlag.
    Stephen George Simpson. with definition 1.2.3 and the discussion following it. For example, taking 90(n) to be the formula n §E Y, we have an instance of comprehension, VYEIXVn(n€X<—>n¢Y), asserting that for any given set Y there exists a ...
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  • The Mean Value Theorem in Second Order Arithmetic.Christopher Hardin & Daniel Velleman - 2001 - Journal of Symbolic Logic 66 (3):1353-1358.
     
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  • Subsystems of Second Order Arithmetic.Stephen G. Simpson - 1999 - Studia Logica 77 (1):129-129.
     
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