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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.
Arithmetical transfinite induction
"... and hierarchies of functions by Zygmunt Ra t a j c z yk (Warszawa) Abstract. We generalize to the case of arithmetical transfinite induction the following three theorems for PA: the Wainer Theorem, the Paris–Harrington Theorem, and a version of the Solovay–Ketonen Theorem. We give uniform proofs us ..."
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and hierarchies of functions by Zygmunt Ra t a j c z yk (Warszawa) Abstract. We generalize to the case of arithmetical transfinite induction the following three theorems for PA: the Wainer Theorem, the Paris–Harrington Theorem, and a version of the Solovay–Ketonen Theorem. We give uniform proofs using combinatorial constructions. 1. Introduction. The