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Well ordering principles and -statements: A pilot study

Journal of Symbolic Logic 86 (2):709-745 (2021)

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An ordinal analysis of parameter free Π12-comprehension.Michael Rathjen - 2005 - Archive for Mathematical Logic 44 (3):263-362.details Abstract.This paper is the second in a series of three culminating in an ordinal analysis of Π12-comprehension. Its objective is to present an ordinal analysis for the subsystem of second order arithmetic with Δ12-comprehension, bar induction and Π12-comprehension for formulae without set parameters. Couched in terms of Kripke-Platek set theory, KP, the latter system corresponds to KPi augmented by the assertion that there exists a stable ordinal, where KPi is KP with an additional axiom stating that every set is contained (...) in an admissible set. (shrink) Download Export citation Bookmark 18 citations
Computable aspects of the Bachmann–Howard principle.Anton Freund - 2019 - Journal of Mathematical Logic 20 (2):2050006.details We have previously established that [Formula: see text]-comprehension is equivalent to the statement that every dilator has a well-founded Bachmann–Howard fixed point, over [Formula: see text]. In this paper, we show that the base theory can be lowered to [Formula: see text]. We also show that the minimal Bachmann–Howard fixed point of a dilator [Formula: see text] can be represented by a notation system [Formula: see text], which is computable relative to [Formula: see text]. The statement that [Formula: see text] (...) Download Export citation Bookmark 5 citations
(1 other version)Set recursion and [product]¹2-logic.J. Girard - 1985 - Annals of Pure and Applied Logic 28 (3):255.details Download Export citation Bookmark 4 citations
What is effective transfinite recursion in reverse mathematics?Anton Freund - 2020 - Mathematical Logic Quarterly 66 (4):479-483.details In the context of reverse mathematics, effective transfinite recursion refers to a principle that allows us to construct sequences of sets by recursion along arbitrary well orders, provided that each set is ‐definable relative to the previous stages of the recursion. It is known that this principle is provable in. In the present note, we argue that a common formulation of effective transfinite recursion is too restrictive. We then propose a more liberal formulation, which appears very natural and is still (...) provable in. (shrink) Download Export citation Bookmark 2 citations
Derivatives of normal functions in reverse mathematics.Anton Freund & Michael Rathjen - 2021 - Annals of Pure and Applied Logic 172 (2):102890.details Download Export citation Bookmark 2 citations
A note on ordinal exponentiation and derivatives of normal functions.Anton Freund - 2020 - Mathematical Logic Quarterly 66 (3):326-335.details Michael Rathjen and the present author have shown that ‐bar induction is equivalent to (a suitable formalization of) the statement that every normal function has a derivative, provably in. In this note we show that the base theory can be weakened to. Our argument makes crucial use of a normal function f with and. We shall also exhibit a normal function g with and. Download Export citation Bookmark 2 citations
Reverse mathematics and ordinal exponentiation.Jeffry L. Hirst - 1994 - Annals of Pure and Applied Logic 66 (1):1-18.details Simpson has claimed that “ATR0 is the weakest set of axioms which permits the development of a decent theory of countable ordinals” [8]. This paper provides empirical support for Simpson's claim. In particular, Cantor's Normal Form Theorem and Sherman's Inequality for countable well-orderings are both equivalent to ATR0. The proofs of these results require a substantial development of ordinal exponentiation and a strengthening of the comparability result in [3]. Download Export citation Bookmark 21 citations
(1 other version)Set recursion and Πhalf-logic.Jean-Yves Girard & Dag Normann - 1985 - Annals of Pure and Applied Logic 28 (3):255-286.details Download Export citation Bookmark 7 citations

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