Ordinal Notations 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.

Takeuti's Well-ordering Proof

G. Genzten’s 1938 proof of the consistency of pure arithmetic was hailed as a success for finitism and constructivism, but his proof requires induction along ordinal notations in Cantor normal form up to the first epsilon number, ε0. This left the task of giving a finitisically acceptable proof of the well-ordering of those ordinal notations, without which Gentzen’s proof could hardly be seen as a success for finitism. In his seminal book Proof Theory G. Takeuti provides such a proof. After a brief philosophical introduction, we provide a reconstruction of Takeuti’s proof including corrections, comments, re-organization and notational adjustments for the sake of clarity. The result is a much longer, but much more tractable proof of the well-ordering of ordinal notations in Cantor normal form less than ε0, that nevertheless follows Takeuti’s strategy closely. We end with some more general comments about that proof strategy and the notion of accessibility more generally.

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.

A HIERARCHY OF COMPUTABLY ENUMERABLE DEGREES

AbstractWe introduce a new hierarchy of computably enumerable degrees. This hierarchy is based on computable ordinal notations measuring complexity of approximation of${\rm{\Delta }}_2^0$functions. The hierarchy unifies and classifies the combinatorics of a number of diverse constructions in computability theory. It does so along the lines of the high degrees (Martin) and the array noncomputable degrees (Downey, Jockusch, and Stob). The hierarchy also gives a number of natural definability results in the c.e. degrees, including a definable antichain.

Decoding Gentzen's Notation

In this note we consider Gentzen's first ordinal notation, used in his first published proof of the consistency of Peano Arithmetic (1936). 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.

Jumps of computably enumerable equivalence relations

We study computably enumerable equivalence relations (or, ceers), under computable reducibility ≤, and the halting jump operation on ceers. We show that every jump is uniform join-irreducible, and thus join-irreducible. Therefore, the uniform join of two incomparable ceers is not equivalent to any jump. On the other hand there exist ceers that are not equivalent to jumps, but are uniform join-irreducible: in fact above any non-universal ceer there is a ceer which is not equivalent to a jump, and is uniform join-irreducible. We also study transfinite iterations of the jump operation. If a is an ordinal notation, and E is a ceer, then let E(a) denote the ceer obtained by transfinitely iterating the jump on E along the path of ordinal notations up to a. In contrast with what happens for the Turing jump and Turing reducibility, where if a set X is an upper bound for the A-arithmetical sets then X(2) computes A(ω), we show that there is a ceer R such that R≥Id(n), for every finite ordinal n, but, for all k, R(k)≱Id(ω) (here Id is the identity equivalence relation). We show that if a,b are notations of the same ordinal less than ω2, then E(a)≡E(b), but there are notations a,b of ω2 such that Id(a) and Id(b) are incomparable. Moreover, there is no non-universal ceer which is an upper bound for all the ceers of the form Id(a) where a is a notation for ω2.

Pure patterns of order 2

We provide mutual elementary recursive order isomorphisms between classical ordinal notations, based on Skolem hulling, and notations from pure elementary patterns of resemblance of order 2, showing that the latter characterize the proof-theoretic ordinal 1∞ of the fragment Π11–CA0 of second order number theory, or equivalently the set theory KPℓ0. As a corollary, we prove that Carlson's result on the well-quasi orderedness of respecting forests of order 2 implies transfinite induction up to the ordinal 1∞. We expect that our approach will facilitate analysis of more powerful systems of patterns.

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.

Sums of Computable Ordinals

We prove that for any computable ordinal α there exists a notation a ∈ O and a partial computable function possessing the following property: if b and c are given notations in the set {t ∈ O | t <O a} for ordinals β and γ respectively and β +γ < α, then this function finds a notation for β+γ in the same set. We construct examples demonstrating that not all notations for ordinals α ≥ ω2 possess this property. Bibliography: 2 titles.

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.

Friedberg numberings in the Ershov hierarchy

We show that for every ordinal notation $${\xi}$$ ? of a nonzero computable ordinal, there exists a $${\Sigma^{-1}_\xi}$$ Σ ? - 1 --computable family which up to equivalence has exactly one Friedberg numbering, which does not induce the least element in the corresponding Rogers semilattice.

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

Rogers semilattices of families of two embedded sets in the Ershov hierarchy

AbstractLet a be a Kleene's ordinal notation of a nonzero computable ordinal. We give a sufficient condition on a, so that for every \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of two embedded sets, i.e., two sets A, B, with A properly contained in B, the Rogers semilattice of the family is infinite. This condition is satisfied by every notation of ω; moreover every nonzero computable ordinal that is not sum of any two smaller ordinals has a notation that satisfies this condition. On the other hand, we also give a sufficient condition on a, that yields that there is a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of two embedded sets, whose Rogers semilattice consists of exactly one element; this condition is satisfied by all notations of every successor ordinal bigger than 1, and by all notations of the ordinal ω + ω; moreover every computable ordinal that is sum of two smaller ordinals has a notation that satisfies this condition. We also show that for every nonzero n ∈ ω, or n = ω, and every notation a of a nonzero ordinal there exists a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of cardinality n, whose Rogers semilattice consists of exactly one element.