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The Unfolding of NonFinitist Arithmetic
, 2000
"... The unfolding of schematic formal systems is a novel concept which was initiated in Feferman [6]. This paper is mainly concerned with the prooftheoretic analysis of various unfolding systems for nonnitist arithmetic NFA. In particular, we examine two restricted unfoldings U 0 (NFA) and U 1 (NFA ..."
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The unfolding of schematic formal systems is a novel concept which was initiated in Feferman [6]. This paper is mainly concerned with the prooftheoretic analysis of various unfolding systems for nonnitist arithmetic NFA. In particular, we examine two restricted unfoldings U 0 (NFA) and U 1 (NFA), as well as a full unfolding, U(NFA). The principal results then state: (i) U 0 (NFA) is equivalent to PA; (ii) U 1 (NFA) is equivalent to RA<! ; (iii) U(NFA) is equivalent to RA< 0 . Thus U(NFA) is prooftheoretically equivalent to predicative analysis.
The Higher Infinite in Proof Theory
 Logic Colloquium '95. Lecture Notes in Logic
, 1995
"... this paper. The exposition here diverges from the presentation given at the conference in two regards. Firstly, the talk began with a broad introduction, explaining the current rationale and goals of ordinaltheoretic proof theory, which take the place of the original Hilbert Program. Since this par ..."
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Cited by 2 (1 self)
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this paper. The exposition here diverges from the presentation given at the conference in two regards. Firstly, the talk began with a broad introduction, explaining the current rationale and goals of ordinaltheoretic proof theory, which take the place of the original Hilbert Program. Since this part of the talk is now incorporated in the first two sections of the BSLpaper [48] there is no point in reproducing it here. Secondly, we shall omit those parts of the talk concerned with infinitary proof systems of ramified set theory as they can also be found in [48] and even more detailed in [45]. Thirdly, thanks to the aforementioned omissions, the advantage of present paper over the talk is to allow for a much more detailed account of the actual information furnished by ordinal analyses and the role of large cardinal hypotheses in devising ordinal representation systems. 2 Observations on ordinal analyses
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.
The strength of Ramsey theorem for coloring relatively large sets
"... We characterize the computational content and the prooftheoretic strength of a Ramseytype theorem for bicolorings of socalled exactly large sets. An exactly large set is a set X ⊂ N such that card(X) = min(X) + 1. The theorem we analyze is as follows. For every infinite subset M of N, for every ..."
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We characterize the computational content and the prooftheoretic strength of a Ramseytype theorem for bicolorings of socalled exactly large sets. An exactly large set is a set X ⊂ N such that card(X) = min(X) + 1. The theorem we analyze is as follows. For every infinite subset M of N, for every coloring C of the exactly large subsets of M in two colors, there exists and infinite subset L of M such that C is constant on all exactly large subsets of L. This theorem is essentially due to Pudlàk and Rödl and independently to Farmaki. We prove that — over Computable Mathematics — this theorem is equivalent to closure under the ω Turing jump (i.e., under arithmetical truth). Natural combinatorial theorems at this level of complexity are rare. Our results give a complete characterization of the theorem from the point of view of Computable Mathematics and of the Proof Theory of Arithmetic. This nicely extends the current knowledge about the strength of Ramsey Theorem. We also show that analogous results hold for a related principle based on the Regressive Ramsey Theorem. In addition we give a further characterization in terms of truth predicates over Peano Arithmetic. We conjecture that analogous results hold for larger ordinals.