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REFLECTION RANKS AND ORDINAL ANALYSIS
Abstract
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 reflection strength order. We prove that there are no descending sequences of sound extensions of in this ordering. Accordingly, we can attach a rank in this order, which we call reflection rank, to any sound extension of . We prove that for any sound theory T extending , the reflection rank of T equals the proof-theoretic ordinal of T . We also prove that the proof-theoretic ordinal of iterated reflection is . Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles.
... Before stating the main result connecting reflection ranks and proof-theoretic ordinals, let me remind the reader that ACA + 0 is the theory ACA 0 + ∀X (X (ω) exists). 4 The following originally appeared as [7,Theorem 5.16]: Theorem 1.2 For 1 1 -sound extensions T of ACA + 0 , the ≺ 1 1 rank of T equals the proof-theoretic ordinal of T . Theorems 1.1 and 1.2 jointly yield the following claims: (i) Sufficiently sound theories can be ranked according to their proof-theoretic strength, given the right notion of proof-theoretic strength. ...
... Hence, Theorem 1.2 connects two distinct topics in proof theory: iterated reflection and ordinal analysis. In [7], the focus was on the iterated reflection side, and Theorem 1.2 was derived using rather complicated proofs of Schmerl-style conservation theorems. In this paper, we present a different proof of Theorem 1.2. ...
... In fact, since the 1 1 truth-definition is available in ACA 0 , this schema follows from a single instance of itself in ACA 0 ; see [7] for discussion. Hence we may regard this schema as a single sentence. ...
There is no infinite sequence of -sound extensions of each of which proves -reflection of the next. This engenders a well-founded “reflection ranking” of -sound extensions of . For any -sound theory T extending , the reflection rank of T equals the proof-theoretic ordinal of T. This provides an alternative characterization of the notion of “proof-theoretic ordinal,” which is one of the central concepts of proof theory. We provide an alternative proof of this theorem using cut-elimination for infinitary derivations.
... The requirement 'S and T being extensions of Σ 1 2 -AC 0 ' is optimal as presented in Example 6.8. Pakhomov and Walsh [14] defined a different notion called reflection rank that also can be used to gauge the strength of theories, which can be thought as an attempt to rank theories under versions of < Con . More precisely, they defined the relation ...
... Reflection rank. [14] defined a different way to gauge the strength of a given theory using Π 1 1 -reflection. More precisely, they defined the following relation: It turns out that ≺ Γ is well-founded if Γ = Π 1 n or Γ = Σ 1 n+1 for n ≥ 1: Theorem 4.4 Theorem 3.2]). ...
... The argument provided in [14] shows ≺Γ Γ is well-founded for Γ = Π 1 n for extensions of Σ 1 n -AC 0 . This allows us to define the ≺Γ Γ -rank for theories extending Σ 1 n -AC 0 for Γ = Π 1 n . ...
The current ordinal analysis provides the proof-theoretic ordinal of a theory, which calibrates the robustness of the -consequences of the theory. We can ask whether there is an ordinal characteristic capturing more complex consequences, and it turns out that we can define the -proof-theoretic ordinal capturing the robustness of the -consequences of a theory. In this paper, we study the behavior of -proof-theoretic ordinal, and it turns out that -proof-theoretic ordinal also follows an analogue of Walsh's characterization of proof-theoretic ordinal.
... Such an approach is not conducive to the goals of this paper. In particular, we will cite a result from [8] that involve quantification over all ordinal notations, where the ordinal notations are presumed to represent arbitrarily large recursive ordinals. Since we are citing this result, we will adopt the approach to ordinal notations from [8] as our official approach. ...
... In particular, we will cite a result from [8] that involve quantification over all ordinal notations, where the ordinal notations are presumed to represent arbitrarily large recursive ordinals. Since we are citing this result, we will adopt the approach to ordinal notations from [8] as our official approach. ...
... In [8] §2.2, a detailed exposition of this approach is given, but we recall the basic points here. We remind the reader that EA is a weak subsystem of first-order arithmetic that includes induction for formulas that are bounded in an exponential term; for details see [3]. ...
It is well-known that natural axiomatic theories are pre-well-ordered by logical strength, according to various characterizations of logical strength such as consistency strength and inclusion of theorems. Though these notions of logical strength coincide for natural theories, they are not generally equivalent. We study analogues of these notions -- such as -reflection strength and inclusion of theorems -- in the presence of an oracle for truths. In this context these notions coincide; moreover, we get genuine pre-well-orderings of axiomatic theories and may drop the non-mathematical quantification over "natural" theories.
... We will explore the analogy between the recursion-theoretic and proof-theoretic well-ordering phenomena throughout this paper. On the one hand, we will describe proof-theoretic theorems from [22,23,24,33] whose statements and proofs were inspired by this recursion-theoretic research. On the other hand, we will also describe purely recursion-theoretic results from [19] that were inspired by the aforementioned proof-theoretic theorems. ...
... Descending sequences. A few theorems concerning descending sequences in proof-theoretic hierarchies appear in [23]. The first such theorem concerns, not the ordering ≺ Con on axiomatic theories, but a closely related structure. ...
... Is there a limit-recursive sequence (T n ) n∈N of Σ 2 -sound extensions of BΣ 1 such that, for each n, T n proves the Σ 2 -soundness of T n+1 ? 8 In [23] this is stated as a result about EA rather than BΣ 1 . This is because we formalize our results in terms of "smooth provability" instead of using the ordinary provability predicate. ...
It is a well-known empirical phenomenon that natural axiomatic theories are pre-well-ordered by consistency strength. Without a precise mathematical definition of "natural," it is unclear how to study this phenomenon mathematically. We will discuss the significance of this problem and survey some strategies that have recently been developed for addressing it. These strategies emphasize the role of reflection principles and ordinal analysis and draw on analogies with research in recursion theory. We will conclude with a discussion of open problems and directions for future research.
... If we consider the second-order arithmetical theory ACA 0 extended with all true Σ 1 1 -sentences, then its provability algebra forms a ◻-founded Magari algebra. This observation can be obtained following the lines of Theorem 3.2 from [11]. ...
... This notion was inspired by an article of Pakhomov and Walsh[11]. 2 This statement was inspired by a correspondence with Tadeusz Litak (see also the proof of Theorem 2.15 from[9]). ...
We examine cyclic, non-well-founded and well-founded derivations in the provability logic . While allowing cyclic derivations does not change the system, the non-well-founded and well-founded derivations we consider define the same proper infinitary extension of . We establish that this extension is strongly algebraic and neighbourhood complete with respect to both local and global semantic consequence relations. In fact, these completeness results are proved for generalizations of global and local consequence relations, which we call global-local. In addition, we prove strong local neighbourhood completeness for the original system (with ordinary derivations only).
... We write |T | RFN to denote the rank of T in this ordering. In [4], Pakhomov and the author give the following definition: ...
... Robust reflection ranks-though defined only for recursively axiomatized theorieswere introduced in [4]. The primary motivation for introducing them came from the widely-noted contrast between consistency strength ordering on all axiomatic theories and the consistency strength ordering on the "natural" axiomatic theories. ...
Ordinal analysis induces a partition of -definable and -sound theories whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence relation is finer than the ordinal analysis partition if both: (1) whenever T and U prove the same sentences; (2) for every set U of true sentences. In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make.
... To see that Theorem 1.4 implies Theorem 1.3, note that if T were a counter-example to Theorem 1.3 then we would get a counter-example to Theorem 1.4 by letting T " T n for each n. Theorem 1.4 extends earlier work [7,8] of Pakhomov and the author, who proved the following: Theorem 1.5 (Pakhomov-W.). There is no sequence pT n q năω of Π 1 1 -sound and Σ 0 1 -definable extensions of ACA 0 such that for each n, T n $ RFN Π 1 1 pT n`1 q. ...
... Pakhomov and the author proved Theorem 4.6 to provide an explanation of the apparent pre-well-ordering of natural theories by proof-theoretic strength; see [16] for a discussion of this phenomenon. In [7,8], Theorem 4.6 is proved using Gödel's second incompleteness theorem. In particular, we show that the theory ACA 0φ , where ϕ states that Theorem 4.6 is false, proves its own consistency. ...
We present an analogue of G\"{o}del's second incompleteness theorem for systems of second-order arithmetic. Whereas G\"{o}del showed that sufficiently strong theories that are -sound and -definable do not prove their own -soundness, we prove that sufficiently strong theories that are -sound and -definable do not prove their own -soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.
... Recent work by James Walsh and others are deeply engaged with proof-theoretic aspects of the natural linearity phenomenon, including proof-theoretic analogues of Martin's Conjecture [27,36,39], and other work on what they refer to as the wellordering phenomenon for natural theories [28,29,38]. In a sense, Walsh's project takes the well-ordering phenomenon as a given starting point, seeking then to answer the question: what is the meaning of "natural" to make it true that natural theories are well-ordered by consistency or interpretability strength? ...
Many set theorists point to the linearity phenomenon in the hierarchy of consistency strength, by which natural theories tend to be linearly ordered and indeed well ordered by consistency strength. Why should it be linear? In this paper I present counterexamples, natural instances of nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength, as natural or as nearly natural as I can make them. I present the cautious enumeration of ZFC and of various large cardinal set theories, which exhibit incomparability and illfoundedness in consistency strength, and yet which, I argue, are natural. I consider the philosophical role played by “natural” in the linearity phenomenon, arguing ultimately that we should abandon empty naturality talk and aim instead to make precise the mathematical and logical features we had found desirable.
... This phenomenon was explored and used for the applications to ordinal analysis [Sch79;Bek03]. We furthermore note that as was shown by the first author and James Walsh [PW21], for systems of second-order arithmetic and their Π 1 1 -consequences, the strength of the systems can be appropriately measured in terms of the length of iterations of Π 1 1 -reflection, which is a measure closely connected with the Π 1 1 -proof theoretic ordinals of theories. ...
... This phenomenon was explored and used for the applications to ordinal analysis [Sch79;Bek03]. We furthermore note that as was shown by the first author and James Walsh [PW21], for systems of second-order arithmetic and their Π 1 1 -consequences, the strength of the systems can be appropriately measured in terms of the length of iterations of Π 1 1 -reflection, which is a measure closely connected with the Π 1 1 -proof theoretic ordinals of theories. ...
Feferman proved in 1962 that any arithmetical theorem is a consequence of a suitable transfinite iteration of full uniform reflection of . This result is commonly known as Feferman's completeness theorem. The purpose of this paper is twofold. On the one hand this is an expository paper, giving two new proofs of Feferman's completeness theorem that, we hope, shed light on this mysterious and often overlooked result. On the other hand, we combine one of our proofs with results from computable structure theory due to Ash and Knight to give sharp bounds on the order types of well-orders necessary to attain the completeness for levels of the arithmetical hierarchy.
... Well-foundedness of ordinals is also heavily related with correctness statements and their relations are widely studied. For the recent developments, see, e.g., [1,31]. ...
... It is a folklore result [1] that this theory is finitely axiomatizable. We need the following folklore lemma, proved, e.g., in [23]: ...
We employ the lens provided by formal truth theory to study axiomatizations of Peano Arithmetic . More specifically, let Elementary Arithmetic be the fragment of , and let be the extension of by the commonly studied axioms of compositional truth . We investigate both local and global properties of the family of first order theories of the form , where is a particular way of expressing “ is true” (using the truth predicate). Our focus is dominantly on two types of axiomatizations, namely: (1) schematic axiomatizations that are deductively equivalent to and (2) axiomatizations that are proof-theoretically equivalent to the canonical axiomatization of .
... Montalbán's result was suggestive because there are other analogies between the equivalence relation ω A 1 " ω B 1 and the equivalence relation |T | WF " |U | WF ; see the author's work with Lutz [4] for details. The characterization of ordinal analysis as an ordering extends earlier work by Pakhomov and the author [6]. They show that, in a large swathe of cases, proof-theoretic ordinals coincide with ranks of theories in a proof-theoretic reflection ordering; this latter ordering is not linear, however, so this earlier work does not yield Theorem 1.10. ...
Ordinal analysis is a research program wherein recursive ordinals are assigned to axiomatic theories. According to conventional wisdom, ordinal analysis measures the strength of theories. Yet what is the attendant notion of strength? In this paper we present abstract characterizations of ordinal analysis that address this question. First, we characterize ordinal analysis as a partition of -definable and -sound theories, namely, the partition whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence relation is finer than the ordinal analysis partition if both: (1) whenever T and U prove the same sentences; (2) for every set U of true sentences. In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make. Second, we characterize ordinal analysis as an ordering on arithmetically-definable and -sound theories, namely, the ordering wherein if the proof-theoretic ordinal of T is less than the proof-theoretic ordinal of U. The standard ways of measuring the strength of theories are consistency strength and inclusion of theorems. We introduce analogues of these notions -- -reflection strength and inclusion of theorems -- in the presence of an oracle for truths, and prove that they coincide with the ordering induced by ordinal analysis.
... Other forms of reflection have also been studied in the literature. Pakhomov and Walsh [18][19][20] studied iterated reflection principles and their proof theoretic ordinals. Pakhomov and Walsh [19] also related ω-model reflection and iterated syntactic reflection. ...
It is known that several variations of the axiom of determinacy play important roles in the study of reverse mathematics, and the relation between the hierarchy of determinacy and comprehension are revealed by Tanaka, Nemoto, Montalb\'an, Shore, and others. We prove variations of a result by Ko{\l}odziejczyk and Michalewski relating determinacy of arbitrary boolean combinations of sets and reflection in second-order arithmetic. Specifically, we prove that: over , - is equivalent to -; -- is equivalent to -; and -- is equivalent to -. We also restate results by Montalb\'an and Shore to show that - is equivalent to - over .
... It is a folklore result [1] that this theory is finitely axiomatizable. We need the following folklore lemma, proved e.g. in [21]: ...
We employ the lens provided by formal truth theory to study axiomatizations of PA (Peano Arithmetic). More specifically, let EA (Elementary Arithmetic) be the fragment I∆0 + Exp of PA, and CT − [EA] be the extension of EA by the commonly studied axioms of compositional truth CT −. We investigate both local and global properties of the family of first order theories of the form CT − [EA] + α, where α is a particular way of expressing "PA is true" (using the truth predicate). Our focus is dominantly on two types of axiomatizations, namely: (1) schematic axiomatizations that are deductively equivalent to PA, and (2) axiomatizations that are proof-theoretically equivalent to the canonical axiomatization of PA.
Many set theorists point to the linearity phenomenon in the hierarchy of consistency strength, by which natural theories tend to be linearly ordered and indeed well ordered by consistency strength. Why should it be linear? In this paper I present counterexamples, natural instances of nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength, as natural or as nearly natural as I can make them. I present diverse cautious enumerations of ZFC and large cardinal set theories, which exhibit incomparability and illfoundedness in consistency strength, and yet, I argue, are natural. I consider the philosophical role played by “natural” in the linearity phenomenon, arguing ultimately that we should abandon empty naturality talk and aim instead to make precise the mathematical and logical features we had found desirable.
We present an analogue of Gödel’s second incompleteness theorem for systems of second-order arithmetic. Whereas Gödel showed that sufficiently strong theories that are -sound and -definable do not prove their own -soundness, we prove that sufficiently strong theories that are -sound and -definable do not prove their own -soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.
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 extending and axiomatizable by a sentence, and for any , where T is augmented with full induction, and denotes the schema of transfinite induction up to for formulas without set parameters.
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 extending and axiomatizable by a sentence, and for any , \begin{align*}T_0+ \mathrm{RFN}_{\varPi^1_{n+2}}(T) \ = \ T_0 + \mathrm{TI}_{\varPi^1_n}(\varepsilon_0), \end{align*}
\begin{align*}T_0+ \mathrm{RFN}_{\varSigma^1_{n+1}}(T) \ = \ T_0+ \mathrm{TI}_{\varPi^1_n}(\varepsilon_0)^{-}, \end{align*}
where T is augmented with full induction, and denotes the schema of transfinite induction up to for formulas without set parameters.
By a well-known result of Kotlarski et al. (1981), first-order Peano arithmetic can be conservatively extended to the theory of a truth predicate satisfying compositional axioms, i.e., axioms stating that the truth predicate is correct on atomic formulae and commutes with all the propositional connectives and quantifiers. This result motivates the general question of determining natural axioms concerning the truth predicate that can be added to while maintaining conservativity over . Our main result shows that conservativity fails even for the extension of obtained by the seemingly weak axiom of disjunctive correctness that asserts that the truth predicate commutes with disjunctions of arbitrary finite size. In particular, implies . Our main result states that the theory coincides with the theory obtained by adding -induction in the language with the truth predicate. This result strengthens earlier work by Kotlarski (1986) and Cieśliński (2010). For our proof we develop a new general form of Visser’s theorem on non-existence of infinite descending chains of truth definitions and prove it by reduction to (Löb’s version of) Gödel’s second incompleteness theorem, rather than by using the Visser–Yablo paradox, as in Visser’s original proof (1989).
We give a general overview of ordinal notation systems arising from reflection calculi, and extend the to represent impredicative ordinals up to those representable in Buchholz's notation.
Progressions of iterated reflection principles can be used as a tool for the ordinal analysis of formal systems. We discuss various notions of proof-theoretic ordinals and compare the information obtained by means of the reflection principles with the results obtained by the more usual proof-theoretic techniques. In some cases we obtain sharper results, e.g., we define proof-theoretic ordinals relevant to logical complexity 10 and, similarly, for any class n0.
We provide a more general version of the fine structure relationships for iterated reflection principles (due to U. Schmerl [25]). This allows us, in a uniform manner, to analyze main fragments of arithmetic axiomatized by restricted forms of induction, including In, In–, In– and their combinations.
We also obtain new conservation results relating the hierarchies of uniform and local reflection principles. In particular, we show that (for a sufficiently broad class of theories T) the uniform 1-reflection principle for T is 2-conservative over the corresponding local reflection principle. This bears some corollaries on the hierarchies of restricted induction schemata in arithmetic and provides a key tool for our generalization of Schmerl's theorem.
We study reflection principles in fragments of Peano arithmetic and their applications to the questions of comparison and classification of arithmetical theories.
Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the tenth publication in the Lecture Notes in Logic series, Per Lindström presents some of the main topics and results in general metamathematics. In addition to standard results of Gödel et al. on incompleteness, (non-)finite axiomatizability, and interpretability, this book contains a thorough treatment of partial conservativity and degrees of interpretability. It comes complete with exercises, and will be useful as a textbook for graduate students with a background in logic, as well as a valuable resource for researchers.
This paper is an investigation of the relationship between Gödel’s second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector that the relation { ( A , B ) ∈ R 2 : O A ≤ H B } \{(A,B) \in \mathbb {R}^2 : \mathcal {O}^A \leq _H B\} is well-founded. We provide an alternative proof of this fact that uses Gödel’s second incompleteness theorem instead of the theory of admissible ordinals. We then derive a semantic version of the second incompleteness theorem, originally due to Mummert and Simpson, from this result. Finally, we turn to the calculation of the ranks of reals in this well-founded relation. We prove that, for any A ∈ R A\in \mathbb {R} , if the rank of A A is α \alpha , then ω 1 A \omega _1^A is the ( 1 + α ) (1 + \alpha ) th admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of X X is ω 1 X \omega _1^X .
Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the third publication in the Perspectives in Logic series, is a much-needed monograph on the metamathematics of first-order arithmetic. The authors pay particular attention to subsystems (fragments) of Peano arithmetic and give the reader a deeper understanding of the role of the axiom schema of induction and of the phenomenon of incompleteness. The reader is only assumed to know the basics of mathematical logic, which are reviewed in the preliminaries. Part I develops parts of mathematics and logic in various fragments. Part II is devoted to incompleteness. Finally, Part III studies systems that have the induction schema restricted to bounded formulas (bounded arithmetic). © 1998 Springer-Verlag Berlin Heidelberg and © 2016 Association for Symbolic Logic under license to Cambridge University Press.
Strictly positive logics recently attracted attention both in the description logic and in the provability logic communities for their combination of efficiency and sufficient expressivity. The language of Reflection Calculus RC consists of implications between formulas built up from propositional variables and constant `true' using only conjunction and diamond modalities which are interpreted in Peano arithmetic as restricted uniform reflection principles. We extend the language of RC by another series of modalities representing the operators associating with a given arithmetical theory T its fragment axiomatized by all theorems of T of arithmetical complexity , for all n>0. We note that such operators, in a precise sense, cannot be represented in the full language of modal logic. We formulate a formal system RC extending RC that is sound and, as we conjecture, complete under this interpretation. We show that in this system one is able to express iterations of reflection principles up to any ordinal . On the other hand, we provide normal forms for its variable-free fragment. Thereby, the variable-free fragment is shown to be decidable and complete w.r.t. its natural arithmetical semantics. Whereas the normal forms for the variable-free formulas of RC correspond in a unique way to ordinals below , the normal forms of RC are more general. It turns out that they are related in a canonical way to the collections of proof-theoretic ordinals of (bounded) arithmetical theories for each complexity level . Finally, we present an algebraic universal model for the variable-free fragment of RC based on Ignatiev's Kripke frame. Our main theorem states the isomorphism of several natural representations of this algebra.
Preliminaries.- A.- I: Arithmetic as Number Theory, Set Theory and Logic.- II: Fragments and Combinatorics.- B.- III: Self-Reference.- IV: Models of Fragments of Arithmetic.- C.- V: Bounded Arithmetic.- Bibliographical Remarks and Further Reading.- Index of Terms.- Index of Symbols.
Foundations of mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. Almost all of the problems studied in this book are motivated by an overriding foundational question: What are the appropriate axioms for mathematics? Through a series of case studies, these axioms are examined to prove particular theorems in core mathematical areas such as algebra, analysis, and topology, focusing on the language of second-order arithmetic, the weakest language rich enough to express and develop the bulk of mathematics. In many cases, if a mathematical theorem is proved from appropriately weak set existence axioms, then the axioms will be logically equivalent to the theorem. Furthermore, only a few specific set existence axioms arise repeatedly in this context, which in turn correspond to classical foundational programs. This is the theme of reverse mathematics, which dominates the first half of the book. The second part focuses on models of these and other subsystems of second-order arithmetic. Additional results are presented in an appendix. © The Association for Symbolic Logic 2009 and Cambridge University Press, 2009.
The fragment of the polymodal provability logic GLP in the language with connectives ┬, Λ, and 〈n〉 for all n ∈ ω is considered. For this fragment, a deductive system is constructed, a Kripke semantics is proposed, and a polynomial bound for the complexity of a decision procedure is obtained.
After a survey of classical predicative proof theory including the computation of the Schütte-Feferman bounding ordinal for predicativity, the book gives an introduction to impredicative proof theory on the examples of axiom systems for non-iterated inductive definitions and Kripke Platex set theory with Pi_0^2 reflection scheme.
In the body of this paper we use the apparatus of mathematical logic to investigate the role of induction in algebraic reasoning. We show that a surprisingly strong form of induction is needed in order to prove certain very basic and simple algebraic lemmas.
This chapter describes a fine structure on hierarchies generated by reflection formulas over primitive recursive arithmetic (PRA) and its extensions. The comparison of the hierarchies nα or the iterated reflection formulas, involving other proof theoretical concepts and the application of the fine structure theorem, yields a lot of results that are much easier to obtain by the fine structure than by their original proofs. This method is a useful tool for dealing with problems concerning consistency and reflection principles, transfinite induction, proofs of restricted complexity, and some other concepts. The usefulness of iterated reflection formulas and the fine structure become evident by comparing the Cn to other proof theoretical concepts and by showing some applications. The chapter discusses induction and iterated reflection formulas, transfinite induction and iterated reflection formulas, fine structure relations for other theories than PRA, and k-consistency and iterated reflection formulas.
We deal with the fragment of modal logic consisting of implications of
formulas built up from the variables and the constant `true' by conjunction and
diamonds only. The weaker language allows one to interpret the diamonds as the
uniform reflection schemata in arithmetic, possibly of unrestricted logical
complexity. We formulate an arithmetically complete calculus with modalities
labeled by natural numbers and \omega, where \omega corresponds to the full
uniform reflection schema, whereas n<\omega corresponds to its restriction to
arithmetical \Pi_{n+1}-formulas. This calculus is shown to be complete w.r.t. a
suitable class of finite Kripke models and to be decidable in polynomial time.
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 ε0 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.
For “natural enough” systems of ordinal notation we show that α times iterated local reflection schema over a sufficiently strong arithmetic T proves the same Π10-sentences as ωα times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exactly at ε-numbers. We also derive the following more general “mixed” formulas estimating the consistency strength of iterated local reflection: for all ordinals α ⩾ 1 and all β, (Tα)β ≡ Π10 Tωα·(1 + β), (Tβ)α ≡ Π10 Tβ + ωα. Here Tα stands for α times iterated local reflection over T, Tβ stands for β times iterated consistency, and ≡ Π10 denotes (provable in T) mutual Π10-conservativity.
The ω-rule,
with the meaning “if the formula A(n) is provable for all n, then the formula ∀xA(x) is provable”, has a certain formal similarity with a uniform reflection principle saying “if A(n) is provable for all n, then ∀xA(x) is true”. There are indeed some hints in the literature that uniform reflection has sometimes been understood as a “formalized ω-rule” (cf. for example S. Feferman [1], G. Kreisel [3], G. H. Müller [7]). This similarity has even another aspect: replacing the induction rule or scheme in Peano arithmetic PA by the ω-rule leads to a complete and sound system PA∞, where each true arithmetical statement is provable. In [2] Feferman showed that an equivalent system can be obtained by erecting on PA a transfinite progression of formal systems PAα based on iterations of the uniform reflection principle according to the following scheme:
Then T = (∪dЄ, PAd, being Kleene's system of ordinal notations, is equivalent to PA∞. Of course, T cannot be an axiomatizable theory.
This paper answers some questions which naturally arise from the Spector-Gandy proof of their theorem that the π 1 ¹ sets of natural numbers are precisely those which are defined by a Σ 1 ¹ formula over the hyperarithmetic sets. Their proof used hierarchies on recursive linear orderings ( H -sets) which are not well orderings. (In this respect they anticipated the study of nonstandard models of set theory.) The proof hinged on the following fact. Let e be a recursive linear ordering. Then e is a well ordering if and only if there is an H -set on e which is hyperarithmetic. It was implicit in their proof that there are recursive linear orderings which are not well orderings, on which there are H -sets. Further information on such nonstandard H -sets (often called pseudohierarchies) can be found in Harrison [4]. It is natural to ask: on which recursive linear orderings are there H -sets?
In Friedman [1] it is shown that there exists a recursive linear ordering e that has no hyperarithmetic descending sequences such that no H -set can be placed on e . In [1] it is also shown that if e is a recursive linear ordering, every point of which has an immediate successor and either has finitely many predecessors or is finitely above a limit point (heretofore called adequate ) such that an H -set can be placed on e , then e has no hyperarithmetic descending sequences. In a related paper, Friedman [2] shows that there is no infinite sequence x n of codes for ω-models of the arithmetic comprehension axiom scheme such that each x n + 1 is a set in the ω-model coded by x n , and each x n +1 is the unique solution of P ( x n , x n +1 ) for some fixed arithmetic P .
Our unexplained notation is that of Rogers [4]. Let P ⊆ 2 N × 2 N . We call a sequence < A n : n ∈ N > of subsets of N a P-sequence iff ∀n(A n +1 = the unique B such that P(A n , B)) .
Theorem. Let P ⊆ 2 N × 2 N be arithmetical. Then there is no P-sequence <A n : n ∈ N> such that ∀n(A′ n +1 ≤ T A n ) .
This theorem improves a result of Friedman [2] who showed that for no arithmetical P is there a P -sequence < A n : n ∈ N > such that A n + 1 is a code for an ω-model of the relative arithmetic comprehension schema, and A n + 1 is present in the model coded by A n , for all n . Other related results are those of Harrison [3], who showed there is a sequence < A n : n ∈ N > such that ∀n<A′ n + 1 ≤ T A n >, and of Enderton and Putnam [1], who showed there is no sequence < A n : n ∈ N > with ∀n(A′ n + 1 ≤ T A n ) and A 0 hyperarithmetic.
Our theorem is closely connected to Gödel's second incompleteness theorem. Its proof is a recursion theoretic parallel to the proof of Gödel's theorem. In §2 we draw a version of Gödel's theorem as a corollary to ours.
Veblen hierarchy in the context of provability algebras, Logic, Methodology and Philosophy of Science
- L Beklemishev
- P Hájek
- L Valdés-Villanueva
- D Westerståhl