Let T(Ω) be the ordinal notation system from Buchholz-Schütte (1988). [The order type of the countable segmentT(Ω)0 is — by Rathjen (1988) — the proof-theoretic ordinal the proof-theoretic ordinal ofACA 0 + (Π 1 l −TR).] In particular let ↦Ω a denote the enumeration function of the infinite cardinals and leta ↦ ψ0 a denote the partial collapsing operation on T(Ω) which maps ordinals of T(Ω) into the countable segment TΩ 0 of T(Ω). Assume that the (fast growing) extended Grzegorczyk (...) hierarchy $(F_a )_{a \in T(\Omega )_0 }$ and the slow growing hierarchy $(G_a )_{a \in T(\Omega )_0 }$ are defined with respect to the natural system of distinguished fundamental sequences of Buchholz and Schütte (1988) in the following way: $$\begin{array}{*{20}c} {G_0 (n): = 0,} & {F_0 (n): = (n + 1)^2 ,} \\ {\begin{array}{*{20}c} {G_{a + 1} (n): = G_a (n) + 1,} \\ {G_l (n): = G_{l[n]} (n),} \\ \end{array} } & {\begin{array}{*{20}c} {F_{a + 1} (n): = \underbrace {F_a (...F_a }_{n + 1 - times}(n)...),} \\ {F_l (n): = F_{l[n]} (n),} \\ \end{array} } \\ \end{array}$$ wherel is a countable limit ordinal (term) and (l[n]) n ∈N is the distinguished fundamental sequence assigned tol. For each ordinal (term)a in T(Ω) and each natural numbern letC n (a) be the formal term which results from the ordinal terma by successively replacing every occurence ofψ a by $\psi _{ - 1 + C_n (a)}$ whereψ −1 is considered as a defined function symbol, namely $\psi _{ - 1} b: = F_{\psi _0 b + 1} (n + 1)$ . (Note thatψ a 0=Ω a ) In this article it is shown that for each ordinal termψ 0 a in T(Ω) there exists a natural numbern 0 such thatψ 0 C n (a) ∈ T(Ω) and $G_{\psi _0 a} (n) \leqslant F_{\psi _0 C_n (a) + 1} (n + 1)$ holds for alln≥n 0. This hierarchy comparison theorem yields a plethora of new results on nontrivial lower bounds for the slow growing ordinals — i.e. ordinals for which the slow growing hierarchy yields a classification of the provably total functions of the theory in question — of various theories of iterated inductive definitions (and subsystems ofKPi) and on the number and size of the subrecursively inaccessible ordinals — i.e. ordinals at which the extended Grzegorczyk hierarchy and the slow growing hierarchy catch up — below the proof-theoretic ordinal ofACA 0+(Π 1 l −TR). In particular these subrecursively inaccessibles ordinals are necessarily of the form $\psi _0 \Omega ..._{\Omega _\omega }$. (shrink)

Inspired by Pohlers' proof-theoretic analysis of KPω we give a straightforward non-metamathematical proof of the (well-known) classification of the provably total functions of $PA, PA + TI(\prec\lceil)$ (where it is assumed that the well-ordering $\prec$ has some reasonable closure properties) and KPω. Our method relies on a new approach to subrecursion due to Buchholz, Cichon and the author.

Inspired by Pohlers' local predicativity approach to Pure Proof Theory and Howard's ordinal analysis of bar recursion of type zero we present a short, technically smooth and constructive strong normalization proof for Gödel's system T of primitive recursive functionals of finite types by constructing an ε 0 -recursive function [] 0 : T → ω so that a reduces to b implies [a] $_0 > [b]_0$ . The construction of [] 0 is based on a careful analysis of the Howard-Schütte (...) treatment of Gödel's T and utilizes the collapsing function ψ: ε 0 → ω which has been developed by the author for a local predicativity style proof-theoretic analysis of PA. The construction of [] 0 is also crucially based on ideas developed in the 1995 paper "A proof of strongly uniform termination for Gödel's T by the method of local predicativity" by the author. The results on complexity bounds for the fragments of T which are obtained in this paper strengthen considerably the results of the 1995 paper. Indeed, for given n let T n be the subsystem of T in which the recursors have type level less than or equal to n+2. (By definition, case distinction functionals for every type are also contained in T n .) As a corollary of the main theorem of this paper we obtain (reobtain?) optimal bounds for the T n -derivation lengths in terms of ω n+2 -descent recursive functions. The derivation lengths of type one functionals from T n (hence also their computational complexities) are classified optimally in terms of $ -descent recursive functions. In particular we obtain (reobtain?) that the derivation lengths function of a type one functional a ∈ T 0 is primitive recursive, thus any type one functional a in T 0 defines a primitive recursive function. Similarly we also obtain (reobtain?) a full classification of T 1 in terms of multiple recursion. As proof-theoretic corollaries we reobtain the classification of the IΣ n+1 -provably recursive functions. Taking advantage from our finitistic and constructive treatment of the terms of Gödel's T we reobtain additionally (without employing continuous cut elimination techniques) that PRA + PRWO ( $\varepsilon_0) \vdash \Pi^0_2$ - Refl(PA) and PRA + PRWO ( $\omega_{n+2}) \vdash \Pi^0_2$ - Refl(I Σ n+1 ), hence PRA + PRWO( $\epsilon_0) \vdash$ Con(PA) and PRA + PRWO( $\omega_{n+2}) \vdash$ Con(IΣ n+1 ). For programmatic reasons we outline in the introduction a vision of how to apply a certain type of infinitary methods to questions of finitary mathematics and recursion theory. We also indicate some connections between ordinals, term rewriting, recursion theory and computational complexity. (shrink)

We give a self-contained and streamlined version of the classification of the provably computable functions of PA. The emphasis is put on illuminating as well as seems possible the intrinsic computational character of the standard cut elimination process. The article is intended to be suitable for teaching purposes and just requires basic familiarity with PA and the ordinals below ε0. (Familiarity with a cut elimination theorem for a Gentzen or Tait calculus is helpful but not presupposed).

Gentzen's three consistency proofs for elementary number theory have a common aim that originates from Hilbert's Program, namely, the aim to justify the application of classical reasoning to quantified propositions in elementary number theory. In addition to this common aim, Gentzen gave a “finitist” interpretation to every number-theoretic proposition with his 1935 and 1936 consistency proofs. In the present paper, we investigate the relationship of this interpretation with intuitionism in terms of the debate between the Hilbert School and the Brouwer (...) School over the significance of consistency proofs. First, we argue that the interpretation had the role of responding to a Brouwer-style objection against the significance of consistency proofs. Second, we propose a way of understanding Gentzen's response to this objection from an intuitionist perspective. (shrink)

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)

The paper contains proof-theoretic investigation on extensions of Kripke-Platek set theory, KP, which accommodate first-order reflection. Ordinal analyses for such theories are obtained by devising cut elimination procedures for infinitary calculi of ramified set theory with Пn reflection rules. This leads to consistency proofs for the theories KP+Пn reflection using a small amount of arithmetic and the well-foundedness of a certain ordinal system with respect to primitive decending sequences. Regarding future work, we intend to avail ourselves of these new cut (...) elimination techniques to attain an ordinal analysis of П12 comprehension by approaching П12 comprehension through transfinite levels of reflection. (shrink)

We present a general schema of easy normalization proofs for finite systems S like first-order arithmetic or subsystems of analysis, which have good infinitary counterparts S ∞ . We consider a new system S ∞ + with essentially the same rules as S ∞ but different derivable objects: a derivation d∈S ∞ + of a sequent Γ contains a derivation Φ∈S of Γ . Three simple conditions on Φ including a normal form theorem for S ∞ + easily imply a (...) weak normalization theorem for S . We give three examples of application of this schema. First, we take S≡PA but restrict the attention to derivations of Σ 1 0 -sentences. In this case it is possible to take S ∞ + to be essentially standard formulation of PA ∞ . Next, we illustrate extension to subsystems of analysis and consider the system BI 1 ∗ of W. Buchholz having the strength of ID 1 , again for derivations of Σ 1 0 -sentences. Finally, we return to the first-order arithmetic to illustrate changes needed to treat derivations of arbitrary formulas. (shrink)

A simple cut elimination proof for arithmetic with the epsilon symbol is used to establish the termination of a modified epsilon substitution process. This opens a possibility of extension to much stronger systems.

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)

Shoenfield’s completeness theorem states that every true first order arithmetical sentence has a recursive \-proof encodable by using recursive applications of the \-rule. For a suitable encoding of Gentzen style \-proofs, we show that Shoenfield’s completeness theorem applies to cut free \-proofs encodable by using primitive recursive applications of the \-rule. We also show that the set of codes of \-proofs, whether it is based on recursive or primitive recursive applications of the \-rule, is \ complete. The same \ completeness (...) results apply to codes of cut free \-proofs. (shrink)

We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory $T_0$ extending $\mathsf {RCA}_0$ and axiomatizable by a $\Pi ^1_{k+2}$ sentence, and for any $n\geq k+1$, $$\begin{align*}T_0+ \mathrm{RFN}_{\varPi^1_{n+2}} \ = \ T_0 + \mathrm{TI}_{\varPi^1_n}, \end{align*}$$ $$\begin{align*}T_0+ \mathrm{RFN}_{\varSigma^1_{n+1}} \ = \ T_0+ \mathrm{TI}_{\varPi^1_n}^{-}, \end{align*}$$ where T is $T_0$ augmented with full induction, and $\mathrm {TI}_{\varPi ^1_n}^{-}$ denotes the schema of transfinite induction up (...) to $\varepsilon _0$ for $\varPi ^1_n$ formulas without set parameters. (shrink)

Let Con↾x denote the finite consistency statement “there are no proofs of contradiction in T with ≤x symbols.” For a large class of natural theories T, Pudlák has shown that the lengths of the shortest proofs of Con↾n in the theory T itself are bounded by a polynomial in n. At the same time he conjectures that T does not have polynomial proofs of the finite consistency statements Con)↾n. In contrast, we show that Peano arithmetic has polynomial proofs of Con)↾n, (...) where Con∗ is the slow consistency statement for Peano arithmetic, introduced by S.-D. Friedman, Rathjen, and Weiermann. We also obtain a new proof of the result that the usual consistency statement Con is equivalent to ε0 iterations of slow consistency. Our argument is proof-theoretic, whereas previous investigations of slow consistency relied on nonstandard models of arithmetic. (shrink)

Schmerl and Beklemishev’s work on iterated reflection achieves two aims: it introduces the important notion of \-ordinal, characterizing the \-theorems of a theory in terms of transfinite iterations of consistency; and it provides an innovative calculus to compute the \-ordinals for a range of theories. The present note demonstrates that these achievements are independent: we read off \-ordinals from a Schütte-style ordinal analysis via infinite proofs, in a direct and transparent way.

We present a new manifestation of Gödel’s second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert’s program. Specifically, we consider a proper extension of Peano arithmetic ( $\mathbf {PA}$ ) by a mathematically meaningful axiom scheme that consists of $\Sigma ^0_2$ -sentences. These sentences assert that each computably enumerable ( $\Sigma ^0_1$ -definable without parameters) property of finite binary trees has a finite basis. Since this fact entails the existence of polynomial time algorithms, it is (...) relevant for computer science. On a technical level, our axiom scheme is a variant of an independence result due to Harvey Friedman. At the same time, the meta-mathematical properties of our axiom scheme distinguish it from most known independence results: Due to its logical complexity, our axiom scheme does not add computational strength. The only known method to establish its independence relies on Gödel’s second incompleteness theorem. In contrast, Gödel’s theorem is not needed for typical examples of $\Pi ^0_2$ -independence (such as the Paris–Harrington principle), since computational strength provides an extensional invariant on the level of $\Pi ^0_2$ -sentences. (shrink)

The goals of reduction andreductionism in the natural sciences are mainly explanatoryin character, while those inmathematics are primarily foundational.In contrast to global reductionistprograms which aim to reduce all ofmathematics to one supposedly ``universal'' system or foundational scheme, reductive proof theory pursues local reductions of one formal system to another which is more justified in some sense. In this direction, two specific rationales have been proposed as aims for reductive proof theory, the constructive consistency-proof rationale and the foundational reduction rationale. However, (...) recent advances in proof theory force one to consider the viability of these rationales. Despite the genuine problems of foundational significance raised by that work, the paper concludes with a defense of reductive proof theory at a minimum as one of the principal means to lay out what rests on what in mathematics. In an extensive appendix to the paper,various reduction relations betweensystems are explained and compared, and arguments against proof-theoretic reduction as a ``good'' reducibilityrelation are taken up and rebutted. (shrink)

Summary. A constructive realizablity interpretation for classical arithmetic is presented, enabling one to extract witnessing terms from proofs of 1 sentences. The interpretation is shown to coincide with modified realizability, under a novel translation of classical logic to intuitionistic logic, followed by the Friedman-Dragalin translation. On the other hand, a natural set of reductions for classical arithmetic is shown to be compatible with the normalization of the realizing term, implying that certain strategies for eliminating cuts and extracting a witness from (...) the proof of a 1 sentence are insensitive to the order in which reductions are applied. (shrink)

In this paper we show that the intuitionistic fixed point theory FiXi over set theories T is a conservative extension of T if T can manipulate finite sequences and has the full foundation schema.

Building on previous work by Mints, Buchholz and Schwichtenberg, a simplified version of continuous normalization for the untyped λ-calculus and Gödel’s is presented and analysed in the coalgebraic framework of non-wellfounded terms with so-called repetition constructors.The primitive recursive normalization function is uniformly continuous w.r.t. the natural metric on non-wellfounded terms. Furthermore, the number of necessary repetition constructors is locally related to the number of reduction steps needed to reach the normal form and its size.It is also shown how continuous normal (...) forms relate to derivations of strong normalizability in the typed λ-calculus and how this leads to new bounds for the sum of the height of the reduction tree and the size of the normal form.Finally, the methods are extended to an infinitary λ-calculus with ω-rule and permutative conversions and this is used to derive a strong form of normalization for an iterative version of Gödel’s system , leading to a value table semantics for number-theoretic functions. (shrink)

Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations. Explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theories can be reobtained in a uniform way.

Gaisi Takeuti extended Gentzen's work to higher-order case in 1950's–1960's and proved the consistency of impredicative subsystems of analysis. He has been chiefly known as a successor of Hilbert's school, but we pointed out in the previous paper that Takeuti's aimed to investigate the relationships between "minds" by carrying out his proof-theoretic project rather than proving the "reliability" of such impredicative subsystems of analysis. Moreover, as briefly explained there, his philosophical ideas can be traced back to Nishida's philosophy in Kyoto's (...) school. For the proving the consistency of such systems, it is crucial to prove the well-foundedness of ordinals called "ordinal diagrams" developed for it. Takeuti presented such arguments several times in order to show that they are admitted in his stand point. As a starting point of investigating his finitist stand point, we formulate the system of ordinal notations up to ε0 and reconstruct the well-foundedness arguments of them. (shrink)