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  1. Which Set Existence Axioms Are Needed to Prove the Separable Hahn-Banach Theorem?Douglas K. Brown & Stephen G. Simpson - 1986 - Annals of Pure and Applied Logic 31:123-144.
  • Riesz Representation Theorem, Borel Measures and Subsystems of Second-Order Arithmetic.Xiaokang Yu - 1993 - Annals of Pure and Applied Logic 59 (1):65-78.
    Yu, X., Riesz representation theorem, Borel measures and subsystems of second-order arithmetic, Annals of Pure and Applied Logic 59 65-78. Formalized concept of finite Borel measures is developed in the language of second-order arithmetic. Formalization of the Riesz representation theorem is proved to be equivalent to arithmetical comprehension. Codes of Borel sets of complete separable metric spaces are defined and proved to be meaningful in the subsystem ATR0. Arithmetical transfinite recursion is enough to prove the measurability of Borel sets for (...)
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  • 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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  • Friedman's Research on Subsystems of Second Order Arithmetic.Stephen G. Simpson - 1990 - Journal of Symbolic Logic 55 (2):870-874.
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