This paper investigates provability and non-provability of well-foundedness of ordinal notations in weak theories of bounded arithmetic. We define a notion of well-foundedness on bounded domains. We show that T21 and S22 can prove the well-foundedness on bounded domains of the ordinal notations below 0 and Γ0. As a corollary, the class of polynomial local search problems, PLS, can be augmented with cost functions that take ordinal values below 0 and Γ0 without increasing the class (...) PLS. (shrink)

This paper investigates provability and non-provability of well-foundedness of ordinal notations in weak theories of bounded arithmetic. We define a notion of well-foundedness on bounded domains. We show that T21 and S22 can prove the well-foundedness on bounded domains of the ordinal notations below 0 and Γ0. As a corollary, the class of polynomial local search problems, PLS, can be augmented with cost functions that take ordinal values below 0 and Γ0 without increasing the class (...) PLS. (shrink)

In this article we investigate whether the following conjecture is true or not: does the addition-free theta functions form a canonical notation system for the linear versions of Friedman’s well-partial-orders with the so-called gap-condition over a finite set of n labels. Rather surprisingly, we can show this is the case for two labels, but not for more than two labels. To this end, we determine the order type of the notation systems for addition-free theta functions in terms of ordinals less (...) than ε0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _0$$\end{document}. We further show that the maximal order type of the Friedman ordering can be obtained by a certain ordinal notation system which is based on specific binary theta functions. (shrink)

We introduce the appropriate iterated version of the system of ordinal notations from [G1] whose order type is the familiar Howard ordinal. As in [G1], our ordinal notations are partly inspired by the ideas from [P] where certain crucial properties of the traditional Munich' ordinal notations are isolated and used in the cut-elimination proofs. As compared to the corresponding “impredicative” Munich' ordinal notations (see e.g. [B1, B2, J, Sch1, Sch2, BSch]), our (...) ordinal notations arearbitrary terms in the appropriate simple term algebra based on the notion of collapsing functions (which we would rather identify as projective functions). In Sect. 1 below we define the systems of ordinal notationsPRJ( ), for any primitive recursive limit wellordering . In Sect. 2 we prove the crucial well-foundness property by using the appropriate well-quasi-ordering property of the corresponding binary labeled trees [G3]. In Sect. 3 we interprete inPRJ( ) the familiar Veblen-Bachmann hierarchy of ordinal functions, and in Sect. 4 we show that the corresponding Buchholz's system of ordinal notationsOT( ) is a proper subsystem ofPRJ( ), although it has the same order type according to [G3] together with the interpretation from Sect. 2 in the terms of labeled trees. In Sect. 5 we use Friedman's approach in order to obtain an appropriate purely combinatorial statement which is not provable in the theory of iterated inductive definitions ID< λ, for arbitrarily large limit ordinalλ. Formal theories, axioms, etc. used below are familiar in the proof theory of subsystems of analysis (see [BFPS, T, BSch]). (shrink)

The notion of a function from ℕ to ℕ defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke–Platek admissible set theory with an axiom of infinity. Formalizing the argument provides (...) an ordinal analysis. (shrink)

We consider five ordinal notation systems of ε0 which are all well-known and of interest in proof-theoretic analysis of Peano arithmetic: Cantor’s system, systems based on binary trees and on countable tree-ordinals, and the systems due to Schütte and Simpson, and to Beklemishev. The main point of this paper is to demonstrate that the systems except the system based on binary trees are equivalent as structured systems, in spite of the fact that they have their origins in different views (...) and trials in proof theory. This is true while Weiermann’s results based on Friedman-style miniaturization indicate that the system based on binary trees is of different character than the others. (shrink)

The standard method of generating countable ordinals from uncountable ordinals can be replaced by a use of fixed point extractors available in the term calculus of Howard’s system. This gives a notion of the intrinsic complexity of an ordinal analogous to the intrinsic complexity of a function described in Gödel’s T.

G. Jäger gave in Arch. Math. Logik Grundlagenforsch. 24 , 49-62, a recursive notation system on a basis of a hierarchy Iαß of α-inaccessible regular ordinals using collapsing functions following W. Buchholz in Ann. Pure Appl. Logic 32 , 195-207. Jäger's system stops, when ordinals α with Iα0 = α enter. This border is now overcome by introducing additional a hierarchy Jαß of weakly inaccessible Mahlo numbers, which is defined similarly to the Jäger hierarchy. An ordinal μ is called (...) Mahlo, if every normal-function f : μ → μ has regular fixpoints. Collapsing is defined for both Mahlo and simply regular ordinals such that for every Mahlo ordinal μ out of the J-hierarchy Ψμα is a regular σ such that Iσ0 = σ. For these regular σ again collapsing functions Ψσ are defined. To get a proper systematical order into the collapsing procedure, a pair of ordinals is associated to σ and α, and the definition of Ψσα is given by recursion on a suitable well-ordering of these pairs. Thus a fairly large system of ordinal notations can be established. It seems rather straightforward, how to extend this setting further. (shrink)

We show the following results on Wainer's notation for a minimal subrecursive inaccessible ordinal τ: First, we give a constructive proof of the collapsing theorem. Secondly, we prove that the slow-growing hierarchy and the fast-growing hierarchy up to τ have elementary properties on increase and domination, which completes Wainer's proof that τ is a minimal subrecursive inaccessible. Our results are obtained by showing a strong normalization theorem for the term structure of the notation. MSC: 03D20, 03F15.

We will introduce a partial ordering $\preceq_1$ on the class of ordinals which will serve as a foundation for an approach to ordinal notations for formal systems of set theory and second-order arithmetic. In this paper we use $\preceq_1$ to provide a new characterization of the ubiquitous ordinal $\epsilon _{0}$.

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 19–77], [G. Wilken, Σ1-Elementarity and Skolem hull operators, Annals of Pure and Applied Logic 145 162–175], and [G. Wilken, Assignment of ordinals to patterns of resemblance, The Journal of Symbolic Logic ]. Within the arithmetical context laid bare in this work, the “-numbers” play a role analogous to the role epsilon numbers play in the ordinal arithmetic based on the notion of Cantor normal form. (shrink)

We define ordinal representations in the simply typed lambda calculus, and consider the ordinal functions representable with respect to these notations. The results of this paper have the same flavor as those of Schwichtenberg and Statman on numeric functions representable in the simply typed lambda calculus. We define four families of ordinal notations; in order of increasing generality of the type of notation, the representable functions consist of the closure under composition of successor and α (...) ωα, addition and α ωα, addition and multiplication, and addition, multiplication, and a “weak” discriminator. (shrink)

It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderedness phenomenon by studying a coarsening of the consistency strength order, namely, the$\Pi ^1_1$reflection strength order. We prove that there are no descending sequences of$\Pi ^1_1$sound extensions of$\mathsf {ACA}_0$in this ordering. Accordingly, we can attach a rank in this order, which we call reflection rank, to any$\Pi (...) ^1_1$sound extension of$\mathsf {ACA}_0$. We prove that for any$\Pi ^1_1$sound theoryTextending$\mathsf {ACA}_0^+$, the reflection rank ofTequals the$\Pi ^1_1$proof-theoretic ordinal ofT. We also prove that the$\Pi ^1_1$proof-theoretic ordinal of$\alpha $iterated$\Pi ^1_1$reflection is$\varepsilon _\alpha $. Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles. (shrink)

This article provides a detailed comparison between two systems of collapsing functions. These functions play a crucial role in proof theory, in the analysis of patterns of resemblance, and the analysis of maximal order types of well partial orders. The exact correspondence given here serves as a starting point for far reaching extensions of current results on patterns and well partial orders. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

In this paper we introduce a recursive notation system O(Π 3 ) of ordinals. An element of the notation system is called an ordinal diagram. The system is designed for proof theoretic study of theories of Π 3 -reflection. We show that for each $\alpha in O(Π 3 ) a set theory KP Π 3 for Π 3 -reflection proves that the initial segment of O(Π 3 ) determined by α is a well ordering. Proof theoretic study for such (...) theories will be reported in [4]. (shrink)

In this note we give a simplified ordinal analysis of first-order reflection. An ordinal notation system$OT$is introduced based on$\psi $-functions. Provable$\Sigma _{1}$-sentences on$L_{\omega _{1}^{CK}}$are bounded through cut-elimination on operator controlled derivations.

In this note we consider Gentzen's first ordinal notation, used in his first published proof of the consistency of Peano Arithmetic. It is a decimal notation, quite different from our current notations. We give a rule to translate this notation into our usual set-theoretic notation and we show some of its peculiarities. Then we indicate how to decode Gentzen's assignment of ordinal notations to derivations and give some examples. Finally, we go through his proof of their (...) decrease after the application of his reduction procedure, giving further examples. (shrink)

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].

In [2] T. J. Carlson introduces an approach to ordinal notation systems which is based on the notion of Σ₁-elementary substructure. We gave a detailed ordinal arithmetical analysis (see [7]) of the ordinal structure based on Σ₁-elementarity as defined in [2]. This involved the development of an appropriate ordinal arithmetic that is based on a system of classical ordinal notations derived from Skolem hull operators, see [6]. In the present paper we establish an effective (...) order isomorphism between the classical and the new system of ordinal notations using the results from [6] and [7]. Moreover, on the basis of a concept of relativization we develop mutual (relatively) elementary recursive assignments which are uniform with respect to the underlying relativization. (shrink)

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.

Buchholz presented a method to build notation systems for infinite sequent-style derivations, analogous to well-known systems of notation for ordinals. The essential feature is that from a notation one can read off by a primitive recursive function its n th predecessor and, e.g. the last rule applied. Here we extend the method to the more general setting of infinite terms, in order to make it applicable in other proof-theoretic contexts as well as in recursion theory. As examples, we use the (...) method to 1. give a new proof of a well-known trade-off theorem , which says that detours through higher types can be eliminated by the use of transfinite recursion along higher ordinals, and 2. construct a continuous normalization operator with an explicit modulus of continuity. (shrink)

In Simmons (Arch Math Logic 43:65–83, 2004), I described a method of producing ordinal notations ‘from below’ (for countable ordinals up to the Howard ordinal) and compared that method with the current popular ‘from above’ method which uses a collapsing function from uncountable ordinals. This ‘from below’ method employs a slight generalization of the normal function—the fruitful functions—and what seems to be a new class of functions—the helpful functions—which exist at all levels of the function space hierarchy (...) over ordinals. Unfortunately, I was rather sparing in my description of these classes of functions. In this paper I am much more generous. I describe the properties of the helpful functions on all finite levels and, in the final section, indicate how they can be used to simplify the generation of ordinal notations. The main aim of this paper is to fill in the details missing from [7]. The secondary aim is to indicate what can be done with helpful functions. Fuller details of this development will appear elsewhere. (shrink)

We suggest an algebraic approach to proof-theoretic analysis based on the notion of graded provability algebra, that is, Lindenbaum boolean algebra of a theory enriched by additional operators which allow for the structure to capture proof-theoretic information. We use this method to analyze Peano arithmetic and show how an ordinal notation system up to 0 can be recovered from the corresponding algebra in a canonical way. This method also establishes links between proof-theoretic ordinal analysis and the work which (...) has been done in the last two decades on provability logic and reflection principles. Because of its abstract algebraic nature, we hope that it will also be of interest for non-prooftheorists. (shrink)

Suppose that α = γ + δ where $\gamma \geq \delta > 0$ . Then there is a graph G = (ω ω α ,E) which has no independent set of order type ω ω α and has no pentagram (a pentagram is a set of five points with all pairs joined by edges). In the notation of Erdos and Rado, who generalized Ramsey's Theorem to this setting, $\omega^{\omega^\alpha} \nrightarrow (\omega^{\omega^\alpha},5)^2.$.

In ordinal analysis of impredicative theories so-called collapsing functions are of central importance. Unfortunately, the definition procedure of these functions makes essential use of uncountable cardinals whereas the notation system that they call into being corresponds to a recursive ordinal. It has long been claimed that, instead, one should manage to develop such functions directly on the basis of admissible ordinals. This paper is meant to show how this can be done. Interpreting the collapsing functions as operating directly (...) on admissible sets also renders a new and perspicuous approach to well-ordering proofs possible. MSC: 03F15, 03F35. (shrink)

Kleene [5] mentions two ways of extending the constructive ordinals. The first is by relativizing the setOof notations for the constructive ordinals, using fundamental sequences which are partial recursive inO. In this way we obtain the setOOwhich provides notations for the ordinals less than ω1O. Continuing the process, the sequenceO,OO,, … and the corresponding ordinalsare obtained. A second possibility is to define higher number classes in which partial recursive functions are used at limit ordinals to provide an “accessibility” (...) mapping from a previously defined number class. The relationship between the ordinals obtained by the two methods of extension has been an open problem. Methods developed in this article are used to show that the two ways of extending the constructive ordinals are equivalent, provided the sets of notations for the higher number classes satisfy certain natural conditions. Equivalence is obtained, not only with respect to ordinals, but also with respect to the forms of the sets of notations for the higher number classes. Specifically, the fundamental fact that the sets of notations for the constructive ordinals are complete Π11sets generalizes to suitably defined higher number classes. As an application we prove that the ordinals of the Addison and Kleene [1] constructive third number class are exactly the ordinals less than ω1Oand the setof notations for their third number class is recursively isomorphic toOO. (shrink)

We introduce and analyze two theories for typed inductive definitions and establish their proof-theoretic ordinal to be the small Veblen ordinal \. We investigate on the one hand the applicative theory \ of functions, inductive definitions, and types. It includes a simple type structure and is a natural generalization of S. Feferman’s system \\). On the other hand, we investigate the arithmetical theory \ of typed inductive definitions, a natural subsystem of \, and carry out a wellordering proof (...) within \ that makes use of fundamental sequences for ordinal notations in an ordinal notation system based on the finitary Veblen functions. The essential properties for describing the ordinal notation system are worked out. (shrink)

In this paper we introduce a notation system for the infinitary derivations occurring in the ordinal analysis of KP + Π₃-Reflection due to Michael Rathjen. This allows a finitary ordinal analysis of KP + Π₃-Reflection. The method used is an extension of techniques developed by Wilfried Buchholz, namely operator controlled notation systems for RS∞-derivations. Similarly to Buchholz we obtain a characterisation of the provably recursive functions of KP + Π₃-Reflection as <-recursive functions where < is the ordering on (...) Rathjen's ordinal notation system J(K). Further we show a conservation result for $\Pi _{2}^{0}$-sentences. (shrink)

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].

In this highly original study, Nuccio Ordine uses the figure of the ass as a lens through which to focus on the thought and writings of the great Renaissance humanist philosopher Giordano Bruno.

“A little masterpiece of originality and clarity.”—George Steiner “A necessary book.”—Roberto Saviano “A wonderful little book that will delight you.”—François Busnel International Best Seller / Now in English for the First Time In this thought-provoking and extremely timely work, Nuccio Ordine convincingly argues for the utility of useless knowledge and against the contemporary fixation on utilitarianism—for the fundamental importance of the liberal arts and against the damage caused by their neglect. Inspired by the reflections of great philosophers and writers (e.g., (...) Plato, Dante, Montaigne, Shakespeare, Borges, and Calvino), Ordine reveals how the obsession for material goods and the cult of utility ultimately wither the spirit, jeopardizing not only schools and universities, art, and creativity, but also our most fundamental values—human dignity, love, and truth. Also included is Abraham Flexner’s 1939 essay “The Usefulness of Useless Knowledge,” which originally prompted Ordine to write this book. Flexner—a founder and the first director of the Institute for Advanced Study in Princeton—offers an impassioned defense of curiosity-driven research and learning. (shrink)