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On Fraïssé’s conjecture for linear orders of finite Hausdorff rank
, 2007
"... We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is ϕ2(0), the first fixed point of the εfunction. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in ACA + 0 + “ϕ2(0) is wellordered ..."
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We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is ϕ2(0), the first fixed point of the εfunction. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in ACA + 0 + “ϕ2(0) is wellordered” and, over RCA0, implies ACA ′ 0 + “ϕ2(0) is wellordered”.
THE VEBLEN FUNCTIONS FOR COMPUTABILITY THEORISTS
, 2010
"... We study the computabilitytheoretic complexity and prooftheoretic strength of the following statements: (1) “If X is a wellordering, then so is εX”, and (2) “If X is a wellordering, then so is ϕ(α, X)”, where α is a fixed computable ordinal and ϕ represents the twoplaced Veblen function. For th ..."
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We study the computabilitytheoretic complexity and prooftheoretic strength of the following statements: (1) “If X is a wellordering, then so is εX”, and (2) “If X is a wellordering, then so is ϕ(α, X)”, where α is a fixed computable ordinal and ϕ represents the twoplaced 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 ACA + 0 over RCA0. To prove the latter statement we need to use ωα iterations of the Turing jump, and we show that the statement is equivalent to Π0 ωαCA0. Our proofs are purely computabilitytheoretic. We also give a new proof of a result of Friedman: the statement “if X is a wellordering, then so is ϕ(X, 0)” is equivalent to ATR0 over RCA0.
Brief introduction to unprovability
"... Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. ..."
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Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. In addition, we obtain a short accessible proof of unprovability of the ParisHarrington Principle. The proof employs old ideas but uses only one colouring and directly extracts the set of indiscernibles from its homogeneous set. We also present modified, abridged statements whose unprovability proofs are especially simple. These proofs were tailored for teaching purposes. The article is intended to be accessible to the widest possible audience of mathematicians, philosophers and computer scientists as a brief survey of the subject, a guide through the literature in the field, an introduction to its modeltheoretic techniques and, finally, a modeltheoretic proof of a modern theorem in the subject. However, some understanding of logic is assumed on the part of the readers. The intended audience of this paper consists of logicians, logicaware mathematicians andthinkers of other backgrounds who are interested in unprovable mathematical statements.