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Reverse mathematics and well-ordering principles

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In S. B. Cooper & Andrea Sorbi (eds.), Computability in Context: Computation and Logic in the Real World. World Scientific (2011)

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  1. Reverse mathematics: the playground of logic.Richard A. Shore - 2010 - Bulletin of Symbolic Logic 16 (3):378-402.
    This paper is essentially the author's Gödel Lecture at the ASL Logic Colloquium '09 in Sofia extended and supplemented by material from some other papers. After a brief description of traditional reverse mathematics, a computational approach to is presented. There are then discussions of some interactions between reverse mathematics and the major branches of mathematical logic in terms of the techniques they supply as well as theorems for analysis. The emphasis here is on ones that lie outside the usual main (...)
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  • The veblen functions for computability theorists.Alberto Marcone & Antonio Montalbán - 2011 - Journal of Symbolic Logic 76 (2):575 - 602.
    We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) "If X is a well-ordering, then so is ε X ", and (2) "If X is a well-ordering, then so is φ(α, X)", where α is a fixed computable ordinal and φ represents the two-placed Veblen function. For the former statement, we show that ω iterations of the Turing jump are necessary in the proof and that the statement is equivalent to ${\mathrm{A}\mathrm{C}\mathrm{A}}_{0}^{+}$ over RCA₀. To prove the (...)
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  • Proof-theoretic strengths of the well-ordering principles.Toshiyasu Arai - 2020 - Archive for Mathematical Logic 59 (3-4):257-275.
    In this note the proof-theoretic ordinal of the well-ordering principle for the normal functions \ on ordinals is shown to be equal to the least fixed point of \. Moreover corrections to the previous paper are made.
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  • Derivatives of normal functions and $$\omega $$ ω -models.Toshiyasu Arai - 2018 - Archive for Mathematical Logic 57 (5-6):649-664.
    In this note the well-ordering principle for the derivative \ of normal functions \ on ordinals is shown to be equivalent to the existence of arbitrarily large countable coded \-models of the well-ordering principle for the function \.
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