We introduce and analyze a new axiomatic theory $\mathsf {CD}$ of truth. The primitive truth predicate can be applied to sentences containing the truth predicate. The theory is thoroughly classical in the sense that $\mathsf {CD}$ is not only formulated in classical logic, but that the axiomatized notion of truth itself is classical: The truth predicate commutes with all quantifiers and connectives, and thus the theory proves that there are no truth value gaps or gluts. To avoid inconsistency, the instances (...) of the T-schema are restricted to determinate sentences. Determinateness is introduced as a further primitive predicate and axiomatized. The semantics and proof theory of $\mathsf {CD}$ are analyzed. (shrink)

Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic systems. Although Gödel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial (...) successes, and generated important advances in logical theory and metatheory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s. (shrink)

This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an (...) introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three “dogmas of proof-theoretic semantics” are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence. (shrink)

In this paper, we describe "metaphysical reductions", in which the well-defined terms and predicates of arbitrary mathematical theories are uniquely interpreted within an axiomatic, metaphysical theory of abstract objects. Once certain (constitutive) facts about a mathematical theory T have been added to the metaphysical theory of objects, theorems of the metaphysical theory yield both an analysis of the reference of the terms and predicates of T and an analysis of the truth of the sentences of T. The well-defined terms and (...) predicates of T are analyzed as denoting abstract objects and abstract relations, respectively, in the background metaphysics, and the sentences of T have a reading on which they are true. After the technical details are sketched, the paper concludes with some observations about the approach. One important observation concerns the fact that the proper axioms of the background theory abstract objects can be reformulated in a way that makes them sound more like logical axioms. Some philosophers have argued that we should accept (something like) them as being logical. (shrink)

In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented (...) in Chapter 2 shows that a completeness proof for propositional logic was found by Hilbert and his assistant Paul Bernays already in 1917--18, and that Bernays's contribution was much greater than is commonly acknowledged. Aside from logic, the main technical contribution of Hilbert's Program are the development of formal mathematical theories and proof-theoretical investigations thereof, in particular, consistency proofs. In this respect Wilhelm Ackermann's 1924 dissertation is a milestone both in the development of the Program and in proof theory in general. Ackermann gives a consistency proof for a second-order version of primitive recursive arithmetic which, surprisingly, explicitly uses a finitistic version of transfinite induction up to www . He also gave a faulty consistency proof for a system of second-order arithmetic based on Hilbert's &egr;-substitution method. Detailed analyses of both proofs in Chapter 3 shed light on the development of finitism and proof theory in the 1920s as practiced in Hilbert's school. ;In a series of papers, Charles Parsons has attempted to map out a notion of mathematical intuition which he also brings to bear on Hilbert's finitism. According to him, mathematical intuition fails to be able to underwrite the kind of intuitive knowledge Hilbert thought was attainable by the finitist. It is argued in Chapter 4 that the extent of finitistic knowledge which intuition can provide is broader than Parsons supposes. According to another influential analysis of finitism due to W. W. Tait, finitist reasoning coincides with primitive recursive reasoning. The acceptance of non-primitive recursive methods in Ackermann's dissertation presented in Chapter 3, together with additional textual evidence presented in Chapter 4, shows that this identification is untenable as far as Hilbert's conception of finitism is concerned. Tait's conception, however, differs from Hilbert's in important respects, yet it is also open to criticisms leading to the conclusion that finitism encompasses more than just primitive recursive reasoning. (shrink)

To the axioms of Peano arithmetic formulated in a language with an additional unary predicate symbol T we add the rules of necessitation and conecessitation T and axioms stating that T commutes with the logical connectives and quantifiers. By a result of McGee this theory is -inconsistent, but it can be approximated by models obtained by a kind of rule-of-revision semantics. Furthermore we prove that FS is equivalent to a system already studied by Friedman and Sheard and give an analysis (...) of its proof theory. (shrink)

The fundamental question in reverse mathematics is to determine which set existence axioms are required to prove particular theorems of mathematics. In addition to being interesting in their own right, answers to this question have consequences in both effective mathematics and the foundations of mathematics. Before discussing these consequences, we need to be more specific about the motivating question.Reverse mathematics is useful for studying theorems of either countable or essentially countable mathematics. Essentially countable mathematics is a vague term that is (...) best explained by an example. Complete separable metric spaces are essentially countable because, although the spaces may be uncountable, they can be understood in terms of a countable basis. Simpson gives the following list of areas which can be analyzed by reverse mathematics: number theory, geometry, calculus, differential equations, real and complex analysis, combinatorics, countable algebra, separable Banach spaces, computability theory, and the topology of complete separable metric spaces. Reverse mathematics is less suited to theorems of abstract functional analysis, abstract set theory, universal algebra, or general topology.Section 2 introduces the major subsystems of second order arithmetic used in reverse mathematics: RCA0, WKL0, ACA0, ATR0 and – CA0. Sections 3 through 7 consider various theorems of ordered group theory from the perspective of reverse mathematics. Among the results considered are theorems on ordered quotient groups, groups and semigroup conditions which imply orderability, the orderability of free groups, Hölder's Theorem, Mal'tsev's classification of the order types of countable ordered groups. (shrink)

Wilfred Sieg. Relative Consistency and Accesible Domains.

Hilbert's finitist program was not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration with Bernays during the period from 1917 to 1922. These notes reveal a dialectic progression from a critical logicism through a radical constructivism toward finitism; the progression has (...) to be seen against the background of the stunning presentation of mathematical logic in the lectures given during the winter term 1917/18. In this paper, I sketch the connection of Hilbert's considerations to issues in the foundations of mathematics during the second half of the 19th century, describe the work that laid the basis of modern mathematical logic, and analyze the first steps in the new subject of proof theory. A revision of the standard view of Hilbert's and Bernays's contributions to the foundational discussion in our century has long been overdue. It is almost scandalous that their carefully worked out notes have not been used yet to understand more accurately the evolution of modern logic in general and of Hilbert's Program in particular. One conclusion will be obvious: the dogmatic formalist Hilbert is a figment of historical (de)construction! Indeed, the study and analysis of these lectures reveal a depth of mathematical-logical achievement and of philosophical reflection that is remarkable. In the course of my presentation many questions are raised and many more can be explored; thus, I hope this paper will stimulate interest for new historical and systematic work. (shrink)

§1. Introduction. The purpose of this paper is, in general, to report the state of the art of ordinal analysis and, in particular, the recent success in obtaining an ordinal analysis for the system of-analysis, which is the subsystem of formal second order arithmetic, Z2, with comprehension confined to-formulae. The same techniques can be used to provide ordinal analyses for theories that are reducible to iterated-comprehension, e.g.,-comprehension. The details will be laid out in [28].Ordinal-theoretic proof theory came into existence in (...) 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. Gentzen fostered hopes that with sufficiently large constructive ordinals one could establish the consistency of analysis, i.e., Z2. Considerable progress has been made in proof theory since Gentzen's tragic death on August 4th, 1945, but an ordinal analysis of Z2is still something to be sought. However, for reasons that cannot be explained here,-comprehension appears to be the main stumbling block on the road to understanding full comprehension, giving hope for an ordinal analysis of Z2in the foreseeable future.Roughly speaking,ordinally informativeproof theory attaches ordinals in a recursive representation system to proofs in a given formal system; transformations on proofs to certain canonical forms are then partially mirrored by operations on the associated ordinals. Among other things, ordinal analysis of a formal system serves to characterize its provably recursive ordinals, functions and functionals and can yield both conservation and combinatorial independence results. (shrink)

Ratajczyk, Z., Subsystems of true arithmetic and hierarchies of functions, Annals of Pure and Applied Logic 64 95–152. The combinatorial method coming from the study of combinatorial sentences independent of PA is developed. Basing on this method we present the detailed analysis of provably recursive functions associated with higher levels of transfinite induction, I, and analyze combinatorial sentences independent of I. Our treatment of combinatorial sentences differs from the one given by McAloon [18] and gives more natural sentences. The same (...) method give also a combinatorial technique with no use of the cut-elimination theorem which is appropriate to study proof-theoretic strength of subsystems of first order arithmetic and some of their expansions. It was used to analyze iterated reflection principle and system of transfinite induction with a satisfaction class. (shrink)

§1. Introduction. The purpose of this paper is, in general, to report the state of the art of ordinal analysis and, in particular, the recent success in obtaining an ordinal analysis for the system of-analysis, which is the subsystem of formal second order arithmetic, Z2, with comprehension confined to-formulae. The same techniques can be used to provide ordinal analyses for theories that are reducible to iterated-comprehension, e.g.,-comprehension. The details will be laid out in [28].Ordinal-theoretic proof theory came into existence in (...) 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. Gentzen fostered hopes that with sufficiently large constructive ordinals one could establish the consistency of analysis, i.e., Z2. Considerable progress has been made in proof theory since Gentzen's tragic death on August 4th, 1945, but an ordinal analysis of Z2is still something to be sought. However, for reasons that cannot be explained here,-comprehension appears to be the main stumbling block on the road to understanding full comprehension, giving hope for an ordinal analysis of Z2in the foreseeable future.Roughly speaking,ordinally informativeproof theory attaches ordinals in a recursive representation system to proofs in a given formal system; transformations on proofs to certain canonical forms are then partially mirrored by operations on the associated ordinals. Among other things, ordinal analysis of a formal system serves to characterize its provably recursive ordinals, functions and functionals and can yield both conservation and combinatorial independence results. (shrink)

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.

§1. Introduction. The purpose of this paper is, in general, to report the state of the art of ordinal analysis and, in particular, the recent success in obtaining an ordinal analysis for the system of -analysis, which is the subsystem of formal second order arithmetic, Z2, with comprehension confined to -formulae. The same techniques can be used to provide ordinal analyses for theories that are reducible to iterated -comprehension, e.g., -comprehension. The details will be laid out in [28].Ordinal-theoretic proof theory (...) came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. Gentzen fostered hopes that with sufficiently large constructive ordinals one could establish the consistency of analysis, i.e., Z2. Considerable progress has been made in proof theory since Gentzen's tragic death on August 4th, 1945, but an ordinal analysis of Z2 is still something to be sought. However, for reasons that cannot be explained here, -comprehension appears to be the main stumbling block on the road to understanding full comprehension, giving hope for an ordinal analysis of Z2 in the foreseeable future.Roughly speaking, ordinally informative proof theory attaches ordinals in a recursive representation system to proofs in a given formal system; transformations on proofs to certain canonical forms are then partially mirrored by operations on the associated ordinals. Among other things, ordinal analysis of a formal system serves to characterize its provably recursive ordinals, functions and functionals and can yield both conservation and combinatorial independence results. (shrink)

§1. Introduction. The purpose of this paper is, in general, to report the state of the art of ordinal analysis and, in particular, the recent success in obtaining an ordinal analysis for the system of-analysis, which is the subsystem of formal second order arithmetic, Z2, with comprehension confined to-formulae. The same techniques can be used to provide ordinal analyses for theories that are reducible to iterated-comprehension, e.g.,-comprehension. The details will be laid out in [28].Ordinal-theoretic proof theory came into existence in (...) 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. Gentzen fostered hopes that with sufficiently large constructive ordinals one could establish the consistency of analysis, i.e., Z2. Considerable progress has been made in proof theory since Gentzen's tragic death on August 4th, 1945, but an ordinal analysis of Z2is still something to be sought. However, for reasons that cannot be explained here,-comprehension appears to be the main stumbling block on the road to understanding full comprehension, giving hope for an ordinal analysis of Z2in the foreseeable future.Roughly speaking,ordinally informativeproof theory attaches ordinals in a recursive representation system to proofs in a given formal system; transformations on proofs to certain canonical forms are then partially mirrored by operations on the associated ordinals. Among other things, ordinal analysis of a formal system serves to characterize its provably recursive ordinals, functions and functionals and can yield both conservation and combinatorial independence results. (shrink)

After sketching the main lines of Hilbert's program, certain well-known andinfluential interpretations of the program are critically evaluated, and analternative interpretation is presented. Finally, some recent developments inlogic related to Hilbert's program are reviewed.

Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though some versions of mathematical instrumentalism (...) are defeated by Gödel’s theorem, not all are. By considering inductive reasons in mathematics, we show that some mathematical instrumentalisms survive the theorem. (shrink)

This paper is an investigation into what could be a goodexplication of ``theory S is reducible to theory T''''. Ipresent an axiomatic approach to reducibility, which is developedmetamathematically and used to evaluate most of the definitionsof ``reducible'''' found in the relevant literature. Among these,relative interpretability turns out to be most convincing as ageneral reducibility concept, proof-theoreticalreducibility being its only serious competitor left. Thisrelation is analyzed in some detail, both from the point of viewof the reducibility axioms and of modal logic.

Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.

This paper is about two topics: 1. systems of explicit mathematics with universes and a non-constructive quantification operator $\mu$; 2. iterated fixed point theories with ordinals. We give a proof-theoretic treatment of both families of theories; in particular, ordinal theories are used to get upper bounds for explicit theories with finitely many universes.

Reductivist programs in logicand philosophy, especially inthe philosophy of mathematics,are reviewed. The paper argues fora ``methodological realism'' towardsnumbers and sets, but still givesreductionism an important place,albeit in methodology/epistemologyrather than in ontology proper.

This paper addresses questions of universality related to ontological engineering, namely aims at substantiating (negative) answers to the following three basic questions: (i) Is there a ‘universal ontology’?, (ii) Is there a ‘universal formal ontology language’?, and (iii) Is there a universally applicable ‘mode of reasoning’ for formal ontologies? To support our answers in a principled way, we present a general framework for the design of formal ontologies resting on two main principles: firstly, we endorse Rudolf Carnap’s principle of logical (...) tolerance by giving central stage to the concept of logical heterogeneity, i.e. the use of a plurality of logical languages within one ontology design. Secondly, to structure and combine heterogeneous ontologies in a semantically well-founded way, we base our work on abstract model theory in the form of institutional semantics, as forcefully put forward by Joseph Goguen and Rod Burstall. In particular, we employ the structuring mechanisms of the heterogeneous algebraic specification language HetCasl for defining a general concept of heterogeneous, distributed, highly modular and structured ontologies, called hyperontologies. Moreover, we distinguish, on a structural and semantic level, several different kinds of combining and aligning heterogeneous ontologies, namely integration, connection, and refinement. We show how the notion of heterogeneous refinement can be used to provide both a general notion of sub-ontology as well as a notion of heterogeneous equivalence of ontologies, and finally sketch how different modes of reasoning over ontologies are related to these different structuring aspects. (shrink)

The so-called weak König's lemma WKL asserts the existence of an infinite path b in any infinite binary tree . Based on this principle one can formulate subsystems of higher-order arithmetic which allow to carry out very substantial parts of classical mathematics but are Π 2 0 -conservative over primitive recursive arithmetic PRA . In Kohlenbach 1239–1273) we established such conservation results relative to finite type extensions PRA ω of PRA . In this setting one can consider also a uniform (...) version UWKL of WKL which asserts the existence of a functional Φ which selects uniformly in a given infinite binary tree f an infinite path Φf of that tree. This uniform version of WKL is of interest in the context of explicit mathematics as developed by S. Feferman. The elimination process in Kohlenbach [10] actually can be used to eliminate even this uniform weak König's lemma provided that PRA ω only has a quantifier-free rule of extensionality QF-ER instead of the full axioms of extensionality for all finite types. In this paper we show that in the presence of , UWKL is much stronger than WKL: whereas WKL remains to be Π 2 0 -conservative over PRA, PRA ω ++ UWKL contains full Peano arithmetic PA. We also investigate the proof–theoretic as well as the computational strength of UWKL relative to the intuitionistic variant of PRA ω both with and without the Markov principle. (shrink)

It is well-known (due to C. Parsons) that the extension of primitive recursive arithmeticPRA by first-order predicate logic and the rule ofΠ 2 0 -inductionΠ 2 0 -IR isΠ 2 0 -conservative overPRA. We show that this is no longer true in the presence of function quantifiers and quantifier-free choice for numbersAC 0,0-qf. More precisely we show that ℐ :=PRA 2 +Π 2 0 -IR+AC 0,0-qf proves the totality of the Ackermann function, wherePRA 2 is the extension ofPRA by number (...) and function quantifiers andΠ 2 0 -IR may contain function parameters. This is true even forPRA 2 +∑ 1 0 -IR+Π 2 0 -IR −+AC 0,0-qf, whereΠ 2 0 -IR − is the restriction ofΠ 2 0 -IR without function parameters. (shrink)

It is well-known (due to C. Parsons) that the extension of primitive recursive arithmeticPRA by first-order predicate logic and the rule ofΠ 2 0 -inductionΠ 2 0 -IR isΠ 2 0 -conservative overPRA. We show that this is no longer true in the presence of function quantifiers and quantifier-free choice for numbersAC 0,0-qf. More precisely we show that ℐ :=PRA 2 +Π 2 0 -IR+AC 0,0-qf proves the totality of the Ackermann function, wherePRA 2 is the extension ofPRA by number (...) and function quantifiers andΠ 2 0 -IR may contain function parameters.This is true even forPRA 2 +∑ 1 0 -IR+Π 2 0 -IR −+AC 0,0-qf, whereΠ 2 0 -IR − is the restriction ofΠ 2 0 -IR without function parameters. (shrink)

In this paper, we study a concept of universe for a truth predicate over applicative theories. A proof-theoretic analysis is given by use of transfinitely iterated fixed point theories . The lower bound is obtained by a syntactical interpretation of these theories. Thus, universes over Frege structures represent a syntactically expressive framework of metapredicative theories in the context of applicative theories.

We give a survey on truth theories for applicative theories. It comprises Frege structures, universes for Frege structures, and a theory of supervaluation. We present the proof-theoretic results for these theories and show their syntactical expressive power. In particular, we present as a novelty a syntactical interpretation of ID1 in a applicative truth theory based on supervaluation.

We give an overview of recent results in ordinal analysis. Therefore, we discuss the different frameworks used in mathematical proof-theory, namely "subsystem of analysis" including "reverse mathematics", "Kripke-Platek set theory", "explicit mathematics", "theories of inductive definitions", "constructive set theory", and "Martin-Löf's type theory".

In this paper we study applicative theories of operations and numbers with the non-constructive minimum operator in the context of a total application operation. We determine the proof-theoretic strength of such theories by relating them to well-known systems like Peano Arithmetic PA and the system <0 of second order arithmetic. Essential use will be made of so-called fixed-point theories with ordinals, certain infinitary term models and Church-Rosser properties.

Hilbert’s program in the philosophy of mathematics comes in two parts. One part is a technical part. To carry out this part of the program one has to prove a certain technical result. The other part of the program is a philosophical part. It is concerned with philosophical questions that are the real aim of the program. To carry out this part one, basically, has to show why the technical part answers the philosophical questions one wanted to have answered. Hilbert (...) probably thought that he had completed the philosophical part of his program, maybe up to a few details. What was left to do was the technical part. To carry it out one, roughly, had to give a precise axiomatization of mathematics and show that it is consistent on purely finitistic grounds. This would come down to giving a relative consistency proof of mathematics in finitist mathematics, or to give a proof-theoretic reduction of mathematics on to finitist mathematics (we will look at these notions in more detail soon). It is widely believed that Gödel’s theorems showed that the technical part of Hilbert’s program could not be carried out. Gödel’s theorems show that the consistency of arithmetic can not even be proven in arithmetic, not to speak of by finitistic means alone. So, the technical part of Hilbert’s program is hopeless, and since Hilbert’s program essentially relied on both the technical and the philosophical part, Hilbert’s program as a whole is hopeless. Justified as this attitude is, it is a bit unfortunate. It is unfortunate because it takes away too much attention from the philosophical part of Hilbert’s program. And this is unfortunate for two reasons. (shrink)

The proof-theoretic results on axiomatic theories oftruth obtained by different authors in recent years are surveyed.In particular, the theories of truth are related to subsystems ofsecond-order analysis. On the basis of these results, thesuitability of axiomatic theories of truth for ontologicalreduction is evaluated.

I survey some important semantical and axiomatic theories of self-referential truth. Kripke's fixed-point theory, the revision theory of truth and appraoches involving fuzzy logic are the main examples of semantical theories. I look at axiomatic theories devised by Cantini, Feferman, Freidman and Sheard. Finally some applications of the theory of self-referential truth are considered.

To the axioms of Peano arithmetic formulated in a language with an additional unary predicate symbol T we add the rules of necessitation φ/Tφ and conecessitation T φ/φ and axioms stating that T commutes with the logical connectives and quantifiers. By a result of McGee this theory is w-inconsistent, but it can be approximated by models obtained by a kind of rule-of-revision semantics. Furthermore we prove that FS is equivalent to a system already studied by Friedman and Sheard and give (...) an analysis of its proof theory. (shrink)

The aim of this paper is the analysis of uniformity in Feferman's explicit mathematics. The proof-strength of those systems for constructive mathematics is determined by reductions to subsystems of second-order arithmetic: If uniformity is absent, the method of standard structures yields that the strength of the join axiom collapses. Systems with uniformity and join are treated via cut elimination and asymmetrical interpretations in standard structures.

A common objection to hierarchical approaches to truth is that they fragment the concept of truth. This paper defends hierarchical approaches in general against the objection of fragmentation. It argues that the fragmentation required is familiar and unprob-lematic, via a comparison with mathematical proof. Furthermore, it offers an explanation of the source and nature of the fragmentation of truth. Fragmentation arises because the concept exhibits a kind of failure of closure under reflection. This paper offers a more precise characterization of (...) the reflection involved, first in the setting of formal theories of truth, and then in a more general setting. (shrink)

The aim of this article is to give the proof-theoretic analysis of various subsystems of Feferman's theory T1 for explicit mathematics which contain the non-constructive μ-operator and join. We make use of standard proof-theoretic techniques such as cut-elimination of appropriate semiformal systems and asymmetrical interpretations in standard structures for explicit mathematics.

The present paper suggests relative truth definability as a tool for comparing conceptual aspects of axiomatic theories of truth and gives an overview of recent developments of axiomatic theories of truth in the light of it. We also show several new proof-theoretic results via relative truth definability including a complete answer to the conjecture raised by Feferman in [13].

This paper is mainly concerned with proof-theoretic analysis of some second-order systems of explicit mathematics with a non-constructive minimum operator. By introducing axioms for variable types we extend our first-order theory BON to the elementary explicit type theory EET and add several forms of induction as well as axioms for μ. The principal results then state: EET plus set induction is proof-theoretically equivalent to Peano arithmetic PA <0).

Feferman, S. and G. Jäger, Systems of explicit mathematics with non-constructive μ-operator. Part I, Annals of Pure and Applied Logic 65 243-263. This paper is mainly concerned with the proof-theoretic analysis of systems of explicit mathematics with a non-constructive minimum operator. We start off from a basic theory BON of operators and numbers and add some principles of set and formula induction on the natural numbers as well as axioms for μ. The principal results then state: BON plus set induction (...) is proof-theoretically equivalent to Peano arithmetic PA; BON plus formula induction is proof-theoretically equivalent to the system <0 of second-order arithmetic. (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)

This paper engages the question ‘Does the consistency of a set of axioms entail the existence of a model in which they are satisfied?’ within the frame of the Frege-Hilbert controversy. The question is related historically to the formulation, proof and reception of Gödel’s Completeness Theorem. Tools from mathematical logic are then used to argue that there are precise senses in which Frege was correct to maintain that demonstrating consistency is as difficult as it can be, but also in which (...) Hilbert was correct to maintain that demonstrating existence given consistency is as easy as it can be. (shrink)

This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program of Friedman and Simpson. We look in particular at: (i) the long arc from Poincar\'e to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak K\"onig's (...) Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others. (shrink)

Proof-theoretic reflection principles are schemas which attempt to express the soundness of arithmetical theories within their own language, e.g., ${\mathtt{{Prov}_{\mathsf {PA}} \rightarrow \varphi }}$ can be understood to assert that any statement provable in Peano arithmetic is true. It has been repeatedly suggested that justification for such principles follows directly from acceptance of an arithmetical theory $\mathsf {T}$ or indirectly in virtue of their derivability in certain truth-theoretic extensions thereof. This paper challenges this consensus by exploring relationships between reflection principles (...) and principles of mathematical and transfinite induction as well as the status of the latter with respect to various foundational characterizations of number theory. (shrink)

ABSTRACT We use Gödel’s incompleteness theorems as a case study for investigating mathematical depth. We examine the philosophical question of what the depth of Gödel’s incompleteness theorems consists in. We focus on the methodological study of the depth of Gödel’s incompleteness theorems, and propose three criteria to account for the depth of the incompleteness theorems: influence, fruitfulness, and unity. Finally, we give some explanations for our account of the depth of Gödel’s incompleteness theorems.

The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theory provides rich opportunities for such analysis. Although the field has its origins in seventeenth century dynamics and nineteenth century statistical mechanics, it employs infinitary, nonconstructive, and structural methods that are characteristically modern. At the same time, computational concerns (...) and recent applications to combinatorics and number theory force us to reconsider the constructive character of the theory and its methods. This paper surveys some recent contributions to the metamathematical study of ergodic theory, focusing on the mean and pointwise ergodic theorems and the Furstenberg structure theorem for measure preserving systems. In particular, I characterize the extent to which these theorems are nonconstructive, and explain how proof-theoretic methods can be used to locate their "constructive content.". (shrink)

This dissertation examines several of the problems that Hilbert discovered in the foundations of mathematics, from a metalogical perspective. The problems manifest themselves in four different aspects of Hilbert’s views: (i) Hilbert’s axiomatic approach to the foundations of mathematics; (ii) His response to criticisms of set theory; (iii) His response to intuitionist criticisms of classical mathematics; (iv) Hilbert’s contribution to the specification of the role of logical inference in mathematical reasoning. This dissertation argues that Hilbert’s axiomatic approach was guided primarily (...) by model theoretical concerns. Accordingly, the ultimate aim of his consistency program was to prove the model-theoretical consistency of mathematical theories. It turns out that for the purpose of carrying out such consistency proofs, a suitable modification of the ordinary first-order logic is needed. To effect this modification, independence-friendly logic is needed as the appropriate conceptual framework. It is then shown how the model theoretical consistency of arithmetic can be proved by using IF logic as its basic logic. Hilbert’s other problems, manifesting themselves as aspects (ii), (iii), and (iv)—most notably the problem of the status of the axiom of choice, the problem of the role of the law of excluded middle, and the problem of giving an elementary account of quantification—can likewise be approached by using the resources of IF logic. It is shown that by means of IF logic one can carry out Hilbertian solutions to all these problems. The two major results concerning aspects (ii), (iii) and (iv) are the following: (a) The axiom of choice is a logical principle; (b) The law of excluded middle divides metamathematical methods into elementary and non-elementary ones. It is argued that these results show that IF logic helps to vindicate Hilbert’s nominalist philosophy of mathematics. On the basis of an elementary approach to logic, which enriches the expressive resources of ordinary first-order logic, this dissertation shows how the different problems that Hilbert discovered in the foundations of mathematics can be solved. (shrink)