Ordinal Notation Systems Research Articles

Bachmann–Howard derivatives

It is generally accepted that H. Friedman’s gap condition is closely related to iterated collapsing functions from ordinal analysis. But what precisely is the connection? We offer the following answer: In a previous paper we have shown that the gap condition arises from an iterative construction on transformations of partial orders. Here we show that the parallel construction for linear orders yields familiar collapsing functions. The iteration step in the linear case is an instance of a general construction that we call ‘Bachmann–Howard derivative’. In the present paper, we focus on the unary case, i.e., on the gap condition for sequences rather than trees and, correspondingly, on addition-free ordinal notation systems. This is partly for convenience, but it also allows us to clarify a phenomenon that is specific to the unary setting: As shown by van der Meeren, Rathjen and Weiermann, the gap condition on sequences admits two linearizations with rather different properties. We will see that these correspond to different recursive constructions of sequences.

Reflection algebras and conservation results for theories of iterated truth

We consider extensions of the language of Peano arithmetic by transfinitely iterated truth definitions satisfying uniform Tarskian biconditionals. Without further axioms, such theories are known to be conservative extensions of the original system of arithmetic. Much stronger systems, however, are obtained by adding either induction axioms or reflection axioms on top of them. Theories of this kind can interpret some well-known predicatively reducible fragments of second-order arithmetic such as iterated arithmetical comprehension.We obtain sharp results on the proof-theoretic strength of these systems using methods of provability logic. Reflection principles naturally define unary operators acting on the semilattice of axiomatizable extensions of our basic theory of iterated truth. The substructure generated by the top element of this algebra provides a canonical ordinal notation system for the class of theories under investigation.Using these notations we obtain conservativity relationships for iterated reflection principles of different logical complexity levels corresponding to the levels of the hyperarithmetical hierarchy, i.e., the analogs of Schmerl's formulas. These relationships, in turn, provide proof-theoretic analysis of our systems and of some related predicatively reducible theories. In particular, we uniformly calculate the ordinals characterizing the standard measures of their proof-theoretic strength, such as provable well-orderings, classes of provably recursive functions, and Π10-ordinals.

Deducibility and independence in Beklemishev's autonomous provability calculus

Beklemishev introduced an ordinal notation system for the Feferman-Schütte ordinal Γ0 based on the autonomous expansion of provability algebras. In this paper we present the logic BC (for Bracket Calculus). The language of BC extends said ordinal notation system to a strictly positive modal language. Thus, unlike other provability logics, BC is based on a self-contained signature that gives rise to an ordinal notation system instead of modalities indexed by some ordinal given a priori. The presented logic is proven to be equivalent to RCΓ0, that is, to the strictly positive fragment of GLPΓ0. We then define a combinatorial statement based on BC and show it to be independent of the theory ATR0 of Arithmetical Transfinite Recursion, a theory of second order arithmetic far more powerful than Peano Arithmetic.

A SIMPLIFIED ORDINAL ANALYSIS OF FIRST-ORDER REFLECTION

AbstractIn 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.

REFLECTION RANKS AND ORDINAL ANALYSIS

AbstractIt 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.

On Quasi Ordinal Diagram Systems

The purposes of this note are the following two; we first generalize Okada-Takeuti's well quasi ordinal diagram theory, utilizing the recent result of Dershowitz-Tzameret's version of tree embedding theorem with gap conditions. Second, we discuss possible use of such strong ordinal notation systems for the purpose of a typical traditional termination proof method for term rewriting systems, especially for second-order (pattern-matching-based) rewriting systems including a rewrite-theoretic version of Buchholz's hydra game.

A flexible type system for the small Veblen ordinal

We introduce and analyze two theories for typed (accessible part) inductive definitions and establish their proof-theoretic ordinal to be the small Veblen ordinal $$\vartheta \Omega ^\omega $$ . We investigate on the one hand the applicative theory $$\mathsf {FIT}$$ of functions, (accessible part) inductive definitions, and types. It includes a simple type structure and is a natural generalization of S. Feferman’s system $$\mathrm {QL}(\mathsf {F}_\mathsf {0}\text {-}\mathsf {IR}_{N})$$ . On the other hand, we investigate the arithmetical theory $$\mathsf {TID}$$ of typed (accessible part) inductive definitions, a natural subsystem of $$\mathsf {ID}_1$$ , and carry out a wellordering proof within $$\mathsf {TID}$$ 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.

Ordinal notation systems corresponding to Friedman’s linearized well-partial-orders with gap-condition

In this article we investigate whether 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 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 0. 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.

On elementary theories of ordinal notation systems based on reflection principles

L.D. Beklemishev has recently introduced a constructive ordinal notation system for the ordinal ɛ 0. We consider this system and its fragments for smaller ordinals ω n (towers of ω-exponentiations of height n). These systems are based on Japaridze’s well-known polymodal provability logic. They are used in the technique of ordinal analysis of the Peano arithmetic PA and its fragments on the basis of iterated reflection schemes. Ordinal notation systems can be regarded as models of the first-order language. We prove that the full notation system and its fragments for ordinals ≥ω 4 have undecidable elementary theories. At the same time, the fragments of the full system for ordinals ≤ω 3 have decidable elementary theories. We also obtain results on decidability of the elementary theory for ordinal notation systems with weaker signatures.

Limitwise monotonic functions relative to the Kleene’s Ordinal Notation System

In this paper we generalize the theorem that previously obtained by the authors for a wider class of linear orders using X-limitwise monotonic functions relative to the Kleene’s Ordinal Notation System.

FRIEDMAN-WEIERMANN STYLE INDEPENDENCE RESULTS BEYOND PEANO ARITHMETIC

We expose a pattern for establishing Friedman-Weiermann style independence results according to which there are thresholds of provability of some parameterized variants of well-partial-ordering. For this purpose, we investigate an ordinal notation system for <TEX>${\vartheta}{\Omega}^{\omega}$</TEX>, the small Veblen ordinal, which is the proof-theoretic ordinal of the theory <TEX>$({\prod}{\frac{1}{2}}-BI)_0$</TEX>. We also show that it sometimes suffices to prove the independence w.r.t. PA in order to obtain the same kind of independence results w.r.t. a stronger theory such as <TEX>$({\prod}{\frac{1}{2}}-BI)_0$</TEX>.

NEW CONSTRUCTIONS OF FUNCTION AND SET CHAINS

Function and set chains can be constructed using ordinal notation systems. Here, earlier results are improved, and longer chains constructed. In particular, a simpler method is given for constructing set chains. AMS Subject Classification: 03E55

Learning with ordinal-bounded memory from positive data

A bounded example memory learner operates incrementally and maintains a memory of finitely many data items. The paradigm is well-studied and known to coincide with set-driven learning. A hierarchy of stronger and stronger learning criteria had earlier been obtained when one considers, for each k∈N, iterative learners that can maintain a memory of at most k previously processed data items. We investigate an extension of the paradigm into the constructive transfinite. For this purpose we use Kleeneʼs universal ordinal notation system O. To each ordinal notation in O one can associate a learning criterion in which the number of times a learner can extend its example memory is bounded by an algorithmic count-down from the notation. We prove a general hierarchy result: if b is larger than a in Kleeneʼs system, then learners that extend their example memory “at most b times” can learn strictly more than learners that can extend their example memory “at most a times”. For notations for ordinals below ω2 the result only depends on the ordinals and is notation-independent. For higher ordinals it is notation-dependent. In the setting of learners with ordinal-bounded memory, we also study the impact of requiring that a learner cannot discard an element from memory without replacing it with a new one. A learner satisfying this condition is called cumulative.

Learning correction grammars

AbstractWe investigate a new paradigm in the context of learning in the limit, namely, learningcorrection grammarsfor classes ofcomputably enumerable (c.e.)languages. Knowing a language may feature a representation of it in terms oftwogrammars. The second grammar is used to make corrections to the first grammar. Such a pair of grammars can be seen as a single description of (or grammar for) the language. We call such grammarscorrection grammars. Correction grammars capture the observable fact that peopledocorrect their linguistic utterances during their usual linguistic activities.We show that learning correction grammars for classes of c.e. languages in theTxtEx-mode(i.e., converging to a single correct correction grammar in the limit) is sometimes more powerful than learning ordinary grammars even in theTxtBc-model (where the learner is allowed to converge to infinitely many syntactically distinct but correct conjectures in the limit). For eachn≥ 0. there is a similar learning advantage, again in learning correction grammars for classes of c.e. languages, but where we compare learning correction grammars that maken+ 1 corrections to those that makencorrections.The concept of a correction grammar can be extended into the constructive transfinite, using the idea of counting-down from notations for transfinite constructive ordinals. This transfinite extension can also be conceptualized as being about learning Ershov-descriptions for c.e. languages. Forua notation in Kleene's general system (O, &lt;o) of ordinal notations for constructive ordinals, we introduce the concept of anu-correction grammar, whereuis used to bound the number of corrections that the grammar is allowed to make. We prove a general hierarchy result: ifuandvare notations for constructive ordinals such thatu&lt;ov. then there are classes of c.e. languages that can beTxtEx-learned by conjecturingv-correction grammars but not by conjecturingu-correction grammars.Surprisingly, we show that—above “ω-many” corrections—it is not possible to strengthen the hierarchy:TxtEx-learningu-correction grammars of classes of c.e. languages, whereuis a notation inOforanyordinal, can be simulated byTxtBc-learningw-correction grammars, wherewis any notation for the smallest infinite ordinalω.

Assignment of ordinals to patterns of resemblance

AbstractIn [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.

A comparison of well-known ordinal notation systems for [formula omitted

We consider five ordinal notation systems of ε 0 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.

A Buchholz derivation system for the ordinal analysis of KP + Π3-reflection

AbstractIn 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 &lt;-recursive functions where &lt; is the ordering on Rathjen's ordinal notation system . Further we show a conservation result for -sentences.

The Bachmann-Howard Structure in Terms of Σ1-Elementarity

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.

Provability algebras and proof-theoretic ordinals, I

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 (syntax-sensitive) information. We use this method to analyze Peano arithmetic and show how an ordinal notation system up to ε 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.

Provability algebras and proof-theoretic ordinals, I

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 (syntax-sensitive) information. We use this method to analyze Peano arithmetic and show how an ordinal notation system up to ε 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.