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We display the entire structure ${\cal R}_2$ coding $\Sigma_1$- and $\Sigma_2$-elementarity on the ordinals. This will enable the analysis of pure $\Sigma_3$-elementary substructures., Comment: Extensive rewrite of the previous version of the paper according to reviewers' recommendations. The paper was corrected, improved, and extended considerably to become readable as stand-alone text. Numerous instructive examples were added. The introduction was extended and a figure added. The preprint is in final form and the corresponding article in press

How do we compare the complexities of various classes of structures? The Turing ordinal of a class of structures, introduced by Jockusch and Soare, is defined in terms of the number of jumps required for coding to be possible. The back-and-forth ordinal, introduced by Montalban, is defined in terms of Σ α -types. The back-and-forth ordinal is (roughly) bounded by the Turing ordinal. In this paper, we show that, if we do not restrict the allowable classes, the reverse inequality need not hold. Indeed, for any computable ordinals α ≤ β we present a class of structures with back-and-forth ordinal α and Turing ordinal β . We also present an example of a class of structures with back-and-forth ordinal 1 but no Turing ordinal.

Each hereditarily finite set is associated with a unique extensional acyclic digraph. Three parameters, indicating the size or richness of a set, are associated with its digraph: the cardinality of the set, and the numbers of nodes and of edges in the digraph. We study the effects on these parameters of the operations of the ordinal arithmetic of sets.

Pohlers's ordinal analysis in his monograph [12] contains some flaws and thereby ends up with incorrect proof-theoretic ordinals of several systems. The present paper determines their correct proof-theoretic ordinals and also supplements [12] with the ordinal analysis of some other relevant impredicative systems.

We describe the countable ordinals in terms of iterations of Mostowski collapsings. This gives a proof-theoretic bound on definable countable ordinals in Zermelo-Fraenkel set theory ZF.

It was recently proved that each subspace of an ordinal space is also orderable. The present note aims to give a simple proof of this fact., peer-reviewed

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 (P). As an application it is shown that whenever KP (P) + AC proves a Π statement then it holds true in the segment V of the von Neumann hierarchy, where τ stands for the Bachmann-Howard ordinal.

A behavioural pseudometric is often defined as the least fixed point of a monotone function F on a complete lattice of 1-bounded pseudometrics. According to Tarski@?s fixed point theorem, this least fixed point can be obtained by (possibly transfinite) iteration of F, starting from the least element @? of the lattice. The smallest ordinal @a such that F^@a(@?)=F^@a^+^1(@?) is known as the closure ordinal of F. We prove that if F is also continuous with respect to the sup-norm, then its closure ordinal is @w. We also show that our result gives rise to simpler and modular proofs that the closure ordinal is @w.

One of the most important operators in soft computing are the triangular norms (t-norms), as well as the combination among them. The ordinal sum of triangular norms on [0, 1] has been used to construct other triangular norms. However, on a bounded lattice, an ordinal sum of t-norms may not be a t-norm. Several necessary and sufficient conditions are presented in this paper for ensuring whether an ordinal sum on a bounded lattice of arbitrary t-norms is, in fact, a t-norm. Moreover, we show that a large set of ordinal sums verify these conditions. Hence, they are very interesting in order to verify whether an ordinal sum on a bounded lattice is a t-norm for a particular family of t-norms.

It is known that an ordinal is the order type of the lexicographic ordering of a regular language if and only if it is less than ωω. We design a polynomial time algorithm that constructs, for each well-ordered regular language L with respect to the lexicographic ordering, given by a deterministic finite automaton, the Cantor Normal Form of its order type. It follows that there is a polynomial time algorithm to decide whether two deterministic finite automata accepting well-ordered regular languages accept isomorphic languages. We also give estimates on the state complexity of the smallest "ordinal automaton" representing an ordinal less than ωω, together with an algorithm that translates each such ordinal to an automaton.

We provide a positive solution for Post’s Problem for ordinal register machines, and also prove that these machines and ordinal Turing machines compute precisely the same partial functions on ordinals. To do so, we construct ordinal register machine programs which compute the necessary functions. In addition, we show that any set of ordinals solving Post’s Problem must be unbounded in the writable ordinals.

The technique of using infinitary rules in an ordinal analysis has been one of the most productive developments in ordinal analysis. Unfortunately, one of the most advanced variants, the Buchholz Ωμ rule, does not apply to systems much stronger than Π11-comprehension. In this paper, we propose a new extension of the Ω rule using game-theoretic quantifiers. We apply this to a system of inductive definitions with at least the strength of a recursively inaccessible ordinal.

Exploiting the fact that Π 1 0 -definable non-monotone inductive definitions have the same closure ordinal as arbitrary arithmetically definable monotone inductive definitions, we show that the proof theoretic ordinal of an axiomatization ( Π 1 0 - FXP ) 0 of Π 1 0 -definable non-monotone inductive definitions coincides with the proof theoretic ordinal of the theory ID 1 of arithmetically definable monotone inductive definitions.

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.

We generalize ordinary register machines on natural numbers to machines whose registers contain arbitrary ordinals. Ordinal register machines are able to compute a recursive bounded truth predicate on the ordinals. The class of sets of ordinals which can be read off the truth predicate satisfies a natural theory SO. SO is the theory of the sets of ordinals in a model of the Zermelo-Fraenkel axioms ZFC. This allows the following characterization of computable sets: a set of ordinals is ordinal register computable if and only if it is an element of Godel’s constructible universe L.

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-elementarily 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.

The exact correspondence between ordinal notations derived from Skolem hull operators, which are classical in ordinal analysis, and descriptions of ordinals in terms of Σ 1 -elementarity, an approach developed by T.J. Carlson, is analyzed in full detail. The ordinal arithmetical tools needed for this purpose were developed in [G. Wilken, Ordinal arithmetic based on Skolem hulling, Annals of Pure and Applied Logic 145 (2) (2007) 130–161]. We show that the least ordinal κ such that κ 1 ∞ (as defined in [T.J. Carlson, Elementary patterns of resemblance, Annals of Pure and Applied Logic 108 (2001) 19–77] and described below) is the proof theoretic ordinal of the set-theoretic system KP l 0 , confirming a claim of Carlson. Moreover, we characterize the class of all ordinals κ such that κ 1 ∞ and provide an ordinal arithmetical analysis of Carlson’s entire structure R 1 in the style of [T.J. Carlson, Ordinal arithmetic and Σ 1 -elementarity, Archive for Mathematical Logic 38 (1999) 449–460].

The notions of finite and infinite second-order characterizability of cardinal and ordinal numbers are developed. Several known results for the case of finite characterizability are extended to infinite characterizability, and investigations of the second-order theory of ordinals lead to some observations about the Fraenkel-Carnap question for well-orders and about the relationship between ordinal characterizability and ordinal arithmetic. The broader significance of cardinal characterizability and the relationships between different notions of characterizability are also discussed.

The Bachmann-Howard structure, that is the segment of ordinal numbers below the proof theoretic ordinal of Kripke-Platek set theory with infinity, is fully characterized in terms of CARLSON’s approach to ordinal notation systems based on the notion of Σ1-elementarity.

Ordinal sums have been introduced in many different contexts, e.g., for posets, semigroups, t-norms, copulas, aggregation operators, or quite recently for hoops. In this contribution, we focus on ordinal sums of t-norms acting on some bounded lattice which is not necessarily a chain or an ordinal sum of posets. Necessary and sufficient conditions are provided for an ordinal sum operation yielding again a t-norm on some bounded lattice whereas the operation is determined by an arbitrary selection of subintervals as carriers for arbitrary summand t-norms. By such also the structure of the underlying bounded lattice is investigated. Further, it is shown that up to trivial cases there are no ordinal sum t-norms on product lattices in general.

It is shown that, according to NF, many of the assertions of ordinal arithmetic involving the T-function which is peculiar to NF turn out to be equivalent to the truth-in-certain-permutation-models of assertions which have perfectly sensible ZF-style meanings, such as: the existence of wellfounded sets of great size or rank, or the nonexistence of small counterexamples to the wellfoundedness of ∈. Everything here holds also for NFU if the permutations are taken to fix all urelemente.

How do ordinals measure the strength and computational power of formal theories? This paper is concerned with the connection between ordinal representation systems and theories established in ordinal analyses. It focusses on results which explain the nature of this connection in terms of semantical and computational notions from model theory, set theory, and generalized recursion theory.

This paper is intended to give for a general mathematical audience (including non-logicians) a survey of intriguing connections between analytic combinatorics and logic. We define the ordinals below e 0 in non-logical terms and we survey a selection of recent results about the analytic combinatorics of these ordinals. Using a versatile and flexible (logarithmic) compression technique we give applications to phase transitions for independence results, Hilbert’s basis theorem, local number theory, Ramsey theory, Hydra games, and Goodstein sequences. We discuss briefly universality and renormalization issues in this context. Finally, we indicate how regularity properties of ordinal count functions can be used to prove logical limit laws.

Termination proofs are of critical importance for establishing the correct behavior of both transformational and reactive computing systems. A general setting for establishing termination proofs involves the use of the ordinal numbers, an extension of the natural numbers into the transfinite that were introduced by Cantor in the nineteenth century and are at the core of modern set theory. We present the first comprehensive treatment of ordinal arithmetic on compact ordinal notations and give efficient algorithms for various operations, including addition, subtraction, multiplication, and exponentiation. Using the ACL2 theorem proving system, we implemented our ordinal arithmetic algorithms, mechanically verified their correctness, and developed a library of theorems that can be used to significantly automate reasoning involving the ordinals. To enable users of the ACL2 system to fully utilize our work required that we modify ACL2, e.g., we replaced the underlying representation of the ordinals and added a large library of definitions and theorems. Our modifications are available starting with ACL2 version 2.8.

This paper deals with a proof theory for a theory T N of \(\Pi _{N}\)-reflecting ordinals using a system \(\mathit{Od}(\Pi _{N})\) of ordinal diagrams. This is a sequel to the previous one (Arai, Ann Pure Appl Log 129:39–92, 2004) in which a theory for \(\Pi _{3}\)-reflecting ordinals is analysed proof-theoretically.

We study separation and covering properties of special subspaces of products of ordinals. In particular, it is proven that certain subspaces of Σ -products of ordinals are quasi-perfect preimages of Σ -products of copies of ω . We obtain as corollaries that products of ordinals are κ -normal and strongly zero-dimensional. Also, σ -products and Σ -products of ordinals are shown to be countably paracompact, κ -normal and strongly zero-dimensional. Normality in Σ -products and σ -products of ordinals is also characterized. It is also shown that any continuous real-valued function on a σ -product of ordinals has countable range.

Proposed was an approach enabling one to reduce studies of the properties of ordinal relations for criteria with an arbitrary (including finite) set of values to similar problems for relations defined over the entire n-dimensional real space Rn. It enabled one to prove that all main findings, methods, and algorithms obtained previously for the ordinal relations on Rn can be extended in arbitrary criterial spaces to the ordinal relations with the criterion scale cardinalities not less than three.

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 Godel’s T.

This paper deals with a proof theory for a theory T22 of recursively Mahlo ordinals in the form of Π2-reflecting on Π2-reflecting ordinals using a subsystem Od(μ) of the system O(μ) of ordinal diagrams in Arai (Arch. Math. Logic 39 (2000) 353). This paper is the first published one in which a proof-theoretic analysis a la Gentzen–Takeuti of recursively large ordinals is expounded.

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 e0 and Γ0. As a corollary, the class of polynomial local search problems, PLS, can be augmented with cost functions that take ordinal values below e0 and Γ0 without increasing the class PLS.

We give several characterizations of ordinal spaces by means of the existence of a continuous selection which has the property that the value of every non-empty closed set is an isolated point of the closed set, and which has some additional properties. This is a generalization of result of Fujii and Nogura [Topology Appl. 91 (1999) 65–69], which characterizes compact ordinal spaces via continuous selections. Furthermore we give another simpler characterization of ordinal spaces via Fell-continuous selections.

Ordinal sums of semigroups in the sense of [3] leading to triangular norms [29] are studied, generalizing the ordinal sum of triangular norms [30]. The summands of these general ordinal sums are fully characterized, and the ordinal irreducibility of triangular norms is investigated, inducing a natural partition of the class of all triangular norms.

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 $\epsilon _{0}$ .

Some basic theorems about ordinal numbers were proved using McCune’s computer program OTTER, building on Quaife’s modification of Godel’s class theory. Our theorems are based on Isbell’s elegant definition of ordinals. Neither the axiom of regularity nor the axiom of choice is used.

This paper uses the framework of reverse mathematics to analyze the proof theoretic content of several statements concerning multiplication of countable well-orderings. In particular, a division algorithm for ordinal arithmetic is shown to be equivalent to the subsystem ATR0.

This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals ∑bT

We investigate proving termination of term rewriting systems by a compositional interpretation of terms in a total well-founded order. This kind of termination is calledtotal termination. Equivalently total termination can be characterized by the existence of an order on ground terms which is total, wellfounded and closed under contexts. For finite signatures, total termination implies simple termination. The converse does not hold. However, total termination generalizes most of the usual techniques for proving termination (including the well-known recursive path order). It turns out that for this kind of termination the only interesting orders below ɛ0 are built from the natural numbers by lexicographic product and the multiset construction. By examples we show that both constructions are essential. Most of the techniques used are based on ordinal arithmetic.

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 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.

G. Jager gave in Arch. Math. Logik Grundlagenforsch. 24 (1984), 49-62, a recursive notation system on a basis of a hierarchy Iαs of α-inaccessible regular ordinals using collapsing functions following W. Buchholz in Ann. Pure Appl. Logic 32 (1986), 195-207. Jager's system stops, when ordinals α with Iα0 = α enter. This border is now overcome by introducing additional a hierarchy Jαs of weakly inaccessible Mahlo numbers, which is defined similarly to the Jager 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.

The class of all ordinal numbers can be partitioned into two subclasses in such a way that neither subclass contains an arithmetic progression of order type ω, where an arithmetic progression of order type τ means an increasing sequence of ordinal numbers (s + δγ)γ r, δ ≠ 0.

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.