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Switch to: References
Citations of:
Ordinal notations based on a weakly Mahlo cardinal
Archive for Mathematical Logic 29 (4):249-263 (1990)
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KPM is a subsystem of set theory designed to formalize a recursively Mahlo universe of sets. In this paper we show that a certain ordinal notation system is sufficient to measure the proof-theoretic strength ofKPM. This involves a detour through an infinitary calculus RS(M), for which we prove several cutelimination theorems. Full cut-elimination is available for derivations of $\Sigma (L_{\omega _1^c } )$ sentences, whereω 1 c denotes the least nonrecursive ordinal. This paper is self-contained, at least from a technical (...) |
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The paper relativizes the method of ordinal analysis developed for Kripke–Platek set theory to theories which have the power set axiom. We show that it is possible to use this technique to extract information about Power Kripke–Platek set theory, KP.As an application it is shown that whenever KP+AC proves a ΠP2 statement then it holds true in the segment Vτ of the von Neumann hierarchy, where τ stands for the Bachmann–Howard ordinal. |
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We define a realizability interpretation of Aczel's Constructive Set Theory CZF into Explicit Mathematics. The final results are that CZF extended by Mahlo principles is realizable in corresponding extensions of T 0 , thus providing relative lower bounds for the proof-theoretic strength of the latter. |
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This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory and Martin-Löf's intuitionistic theory of types. This paper treats Mahlo's π-numbers which give rise classically to the enumerations of inaccessibles of all transfinite orders. We extend the axioms of CZF and show that the resulting theory, when augmented by the tertium non-datur, is equivalent to ZF plus (...) |
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In the first part we show why ordinals and ordinal notations are naturally connected with proof theoretical research. We introduce the program of ordinal analysis. The second part gives examples of applications of ordinal analysis. |
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Induction–recursion is a powerful definition method in intuitionistic type theory. It extends inductive definitions and allows us to define all standard sets of Martin-Löf type theory as well as a large collection of commonly occurring inductive data structures. It also includes a variety of universes which are constructive analogues of inaccessibles and other large cardinals below the first Mahlo cardinal. In this article we give a new compact formalization of inductive–recursive definitions by modeling them as initial algebras in slice categories. (...) |
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In this paper, we develop primitive recursive analogues of regular cardinals by using ordinal representation systems for KPi and KPM. We also define primitive recursive analogues of inaccessible and hyperinaccessible cardinals. Moreover, we characterize the primitive recursive analogue of the least (uncountable) regular cardinal. |
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In this article we show how to extract with the use of the Buchholz -Cichon-Weiermann approach to subrecursive hierarchies from Rathjen's 1991 ordinal analysis of KPM a characterization of the provably total number-theoretic functions of KPM and some of its subsystems in a uniform and direct way. |
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Taking up ordinal notations derived from Skolem hull operators familiar in the field of infinitary proof theory we develop a toolkit of ordinal arithmetic that generally applies whenever ordinal structures are analyzed whose combinatorial complexity does not exceed the strength of the system of set theory. The original purpose of doing so was inspired by the analysis of ordinal structures based on elementarity invented by T.J. Carlson, see [T.J. Carlson, Elementary patterns of resemblance, Annals of Pure and Applied Logic 108 (...) |
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It is shown how the strong ordinal notation systems that figure in proof theory and have been previously defined by employing large cardinals, can be developed directly on the basis of their recursively large counterparts. Thereby we provide a completely new approach to well-ordering proofs as will be exemplified by determining the proof-theoretic ordinal of the systemKPM of [R91]. |
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In this paper we introduce a recursive notation system $O(\mu)$ of ordinals. An element of the notation system is called an ordinal diagram following G. Takeuti [25]. The system is designed for proof theoretic study of theories of recursively Mahlo universes. We show that for each $\alpha<\Omega$ in $O(\mu)$ KPM proves that the initial segment of $O(\mu)$ determined by $\alpha$ is a well ordering. Proof theoretic study for such theories will be reported in [9]. |
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We study the logical and computational properties of basic theorems of uncountable mathematics, including the Cousin and Lindelöf lemma published in 1895 and 1903. Historically, these lemmas were among the first formulations of open-cover compactness and the Lindelöf property, respectively. These notions are of great conceptual importance: the former is commonly viewed as a way of treating uncountable sets like e.g. [Formula: see text] as “almost finite”, while the latter allows one to treat uncountable sets like e.g. [Formula: see text] (...) |
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In this article we define a new and transparent concept of total collapsing functions for an ordinal notation system which is characteristic for the theory (Δ 2 1 -CA)+(BI). We show that our construction allows the application of Pohler's method of local predicativity as presented in [2] which yields a perspicious proof-theoretic analysis of (Δ 2 1 -CA)+(BI) being not much more complicated than for ID1. |
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