This paper provides an alternative proof of the Gibbard–Satterthwaite theorem.

We show that the consistency of the first order arithmetic $PA$ follows from the pointwise induction up to the Howard ordinal. Our proof differs from U. Schmerl [Sc]: We do not need Girard's Hierarchy Comparison Theorem. A modification on the ordinal assignment to proofs by Gentzen and Takeuti [T] is made so that one step reduction on proofs exactly corresponds to the stepping down $\alpha\mapsto\alpha [1]$ in ordinals. Also a generalization to theories $ID_q$ of finitely iterated inductive (...) definitions is proved. (shrink)

In this paper, we prove the wellfoundedness of recursive notation systems for reflecting ordinals up to Π₃-reflection by relevant inductive definitions.

Philosophers who regard some mathematical proofs as explaining why theorems hold, and others as merely proving that they do hold, disagree sharply about the explanatory value of proofs by mathematical induction. I offer an argument that aims to resolve this conflict of intuitions without making any controversial presuppositions about what mathematical explanations would be.

1. Pohlers and The Problem. I first met Wolfram Pohlers at a workshop on proof theory organized by Walter Felscher that was held in Tübingen in early April, 1973. Among others at that workshop relevant to the work surveyed here were Kurt Schütte, Wolfram’s teacher in Munich, and Wolfram’s fellow student Wilfried Buchholz. This is not meant to slight in the least the many other fine logicians who participated there.2 In Tübingen I gave a couple of survey lectures on results (...) and problems in proof theory that had been occupying much of my attention during the previous decade. The following was the central problem that I emphasized there: The need for an ordinally informative, conceptually clear, proof-theoretic reduction of classical theories of iterated arithmetical inductive definitions to corresponding constructive systems. As will be explained below, meeting that need would be significant for the then ongoing efforts at establishing the constructive foundation for and proof-theoretic ordinal analysis of certain impredicative subsystems of classical analysis. I also spoke in Tübingen about. (shrink)

Does the past rationally bear on the future? David Hume argued that we lack good reason to think that it does. He insisted in particular that we lack — and forever will lack — anything like a demonstrative proof of such a rational bearing. A surprising mathematical result can be read as an invitation to reconsider Hume's confidence.

We characterize the proof-theoretic strength of systems of explicit mathematics with a general principle asserting the existence of least fixed points for monotone inductive definitions, in terms of certain systems of analysis and set theory. In the case of analysis, these are systems which contain the Σ12-axiom of choice and Π12-comprehension for formulas without set parameters. In the case of set theory, these are systems containing the Kripke-Platek axioms for a recursively inaccessible universe together with the existence of a (...) stable ordinal. In all cases, the exact strength depends on what forms of induction are admitted in the respective systems. (shrink)

The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs. Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techniques that mathematicians adopt to prove their results, and offers advice and strategies for constructing proofs. It will improve students’ ability to understand proofs and construct correct proofs (...) of their own. The first chapter of the text introduces the kind of reasoning that mathematicians use when writing their proofs and gives some example proofs to set the scene. The book then describes basic logic to enable an understanding of the structure of both individual mathematical statements and whole mathematical proofs. It also explains the notions of sets and functions and dissects several proofs with a view to exposing some of the underlying features common to most mathematical proofs. The remainder of the book delves further into different types of proof, including direct proof, proof using contrapositive, proof by contradiction, and mathematical induction. The authors also discuss existence and uniqueness proofs and the role of counter examples. (shrink)

In a previous paper, an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction has been demonstrated if the case for “n = 3” is granted as proved only arithmetically (which is a fact a long time ago), furthermore in a way accessible to Fermat himself though without being absolutely and precisely correct. The present paper elucidates the contemporary mathematical background, from which an inductive proof of FLT can be inferred since its proof for the case for (...) “n = 3” has been known for a long time. It needs “Hilbert mathematics”, which is inherently complete unlike the usual “Gödel mathematics”, and based on “Hilbert arithmetic” to generalize Peano arithmetic in a way to unify it with the qubit Hilbert space of quantum information. An “epoché to infinity” (similar to Husserl’s “epoché to reality”) is necessary to map Hilbert arithmetic into Peano arithmetic in order to be relevant to Fermat’s age. Furthermore, the two linked semigroups originating from addition and multiplication and from the Peano axioms in the final analysis can be postulated algebraically as independent of each other in a “Hamilton” modification of arithmetic supposedly equivalent to Peano arithmetic. The inductive proof of FLT can be deduced absolutely precisely in that Hamilton arithmetic and the pransfered as a corollary in the standard Peano arithmetic furthermore in a way accessible in Fermat’s epoch and thus, to himself in principle. A future, second part of the paper is outlined, getting directed to an eventual proof of the case “n=3” based on the qubit Hilbert space and the Kochen-Specker theorem inferable from it. (shrink)

The paper is a continuation of another paper published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The (...) relevant mathematical structure is Hilbert arithmetic in a wide sense, in the framework of which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual to each other. A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction. If the former is the case, the application of set theory and arithmetic can remain disjunctively divided: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. If the latter is the case, the proof is thoroughly within set theory. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided in both cases. The idea of “arithmetic mechanics” is sketched: it might deduce the basic physical dimensions of mechanics (mass, time, distance) from the axioms of arithmetic after a relevant generalization, Furthermore, a future Part III of the paper is suggested: FLT by mediation of Hilbert arithmetic in a wide sense can be considered as another expression of Gleason’s theorem in quantum mechanics: the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information. The availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent to the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem. (shrink)

I set up two axiomatic theories of inductive support within the framework of Kolmogorovian probability theory. I call these theories ‘Popperian theories of inductive support’ because I think that their specific axioms express the core meaning of the word ‘inductive support’ as used by Popper (and, presumably, by many others, including some inductivists). As is to be expected from Popperian theories of inductive support, the main theorem of each of them is an anti-induction theorem, the stronger (...) one of them saying, in fact, that the relation of inductive support is identical with the empty relation. It seems to me that an axiomatic treatment of the idea(s) of inductive support within orthodox probability theory could be worthwhile for at least three reasons. Firstly, an axiomatic treatment demands from the builder of a theory of inductive support to state clearly in the form of specific axioms what he means by ‘inductive support’. Perhaps the discussion of the new anti-induction proofs of Karl Popper and David Miller would have been more fruitful if they had given an explicit definition of what inductive support is or should be. Secondly, an axiomatic treatment of the idea(s) of inductive support within Kolmogorovian probability theory might be accommodating to those philosophers who do not completely trust Popperian probability theory for having theorems which orthodox Kolmogorovian probability theory lacks; a transparent derivation of anti-induction theorems within a Kolmogorovian frame might bring additional persuasive power to the original anti-induction proofs of Popper and Miller, developed within the framework of Popperian probability theory. Thirdly, one of the main advantages of the axiomatic method is that it facilitates criticism of its products: the axiomatic theories. On the one hand, it is much easier than usual to check whether those statements which have been distinguished as theorems really are theorems of the theory under examination. On the other hand, after we have convinced ourselves that these statements are indeed theorems, we can take a critical look at the axioms—especially if we have a negative attitude towards one of the theorems. Since anti-induction theorems are not popular at all, the adequacy of some of the axioms they are derived from will certainly be doubted. If doubt should lead to a search for alternative axioms, sheer negative attitudes might develop into constructive criticism and even lead to new discoveries. -/- I proceed as follows. In section 1, I start with a small but sufficiently strong axiomatic theory of deductive dependence, closely following Popper and Miller (1987). In section 2, I extend that starting theory to an elementary Kolmogorovian theory of unconditional probability, which I extend, in section 3, to an elementary Kolmogorovian theory of conditional probability, which in its turn gets extended, in section 4, to a standard theory of probabilistic dependence, which also gets extended, in section 5, to a standard theory of probabilistic support, the main theorem of which will be a theorem about the incompatibility of probabilistic support and deductive independence. In section 6, I extend the theory of probabilistic support to a weak Popperian theory of inductive support, which I extend, in section 7, to a strong Popperian theory of inductive support. In section 8, I reconsider Popper's anti-inductivist theses in the light of the anti-induction theorems. I conclude the paper with a short discussion of possible objections to our anti-induction theorems, paying special attention to the topic of deductive relevance, which has so far been neglected in the discussion of the anti-induction proofs of Popper and Miller. (shrink)

The basic idea by means of which Popper and Miller proved the non-existence of inductive probabilistic support in 1983/1985/1987, is used to prove that inductive probabilistic countersupport does exist. So it seems that after falsification has won over verification on the deductive side of science, countersupport wins over support on the inductive side.

The method of mathematical induction: The method of mathematical induction -- Examples and exercises -- The proof of induction of some theorems of elemetary algebra -- Solutions.

This paper explores some of the benefits informal logic may have for the analysis of mathematical inference. It shows how Stephen Toulmin’s pioneering treatment of defeasible argumentation may be extended to cover the more complex structure of mathematical proof. Several common proof techniques are represented, including induction, proof by cases, and proof by contradiction. Affinities between the resulting system and Imre Lakatos’s discussion of mathematical proof are then explored.

Various forms of mathematical induction are applicable to domains with some kinds of order. This naturally leads to the questions about the possibility of unification of different inductions and their generalization to wider classes of ordered domains. In the paper we propose a common framework for formulating induction proof principles in various structures and apply it to partially ordered sets. In this framework we propose a fixed induction principle which is indirectly applicable to the class of all posets. In a (...) certain sense, this result provides a solution to the problem of formulating a common generalization of a number of well-known induction principles for discrete and continuous ordered structures. (shrink)

Bar Induction occupies a central place in Brouwerian mathematics. This note is concerned with the strength of Bar Induction on the basis of Constructive Zermelo-Fraenkel Set Theory, CZF. It is shown that CZF augmented by decidable Bar Induction proves the 1-consistency of CZF. This answers a question of P. Aczel who used Bar Induction to give a proof of the Lusin Separation Theorem in the constructive set theory CZF.

Handbook of Mathematical Induction: Theory and Applications shows how to find and write proofs via mathematical induction. This comprehensive book covers the theory, the structure of the written proof, all standard exercises, and hundreds of application examples from nearly every area of mathematics. In the first part of the book, the author discusses different inductive techniques, including well-ordered sets, basic mathematical induction, strong induction, double induction, infinite descent, downward induction, and several variants. He then introduces ordinals and cardinals, (...) transfinite induction, the axiom of choice, Zorn’s lemma, empirical induction, and fallacies and induction. He also explains how to write inductive proofs. The next part contains more than 750 exercises that highlight the levels of difficulty of an inductive proof, the variety of inductive techniques available, and the scope of results provable by mathematical induction. Each self-contained chapter in this section includes the necessary definitions, theory, and notation and covers a range of theorems and problems, from fundamental to very specialized. The final part presents either solutions or hints to the exercises. Slightly longer than what is found in most texts, these solutions provide complete details for every step of the problem-solving process. (shrink)

In proof-theoretic semantics, model-theoretic validity is replaced by proof-theoretic validity. Validity of formulae is defined inductively from a base giving the validity of atoms using inductive clauses derived from proof-theoretic rules. A key aim is to show completeness of the proof rules without any requirement for formal models. Establishing this for propositional intuitionistic logic raises some technical and conceptual issues. We relate Sandqvist’s (complete) base-extension semantics of intuitionistic propositional logic to categorical proof theory in presheaves, reconstructing categorically the soundness (...) and completeness arguments, thereby demonstrating the naturality of Sandqvist’s constructions. This naturality includes Sandqvist’s treatment of disjunction that is based on its second-order or elimination-rule presentation. These constructions embody not just validity, but certain forms of objects of justifications. This analysis is taken a step further by showing that from the perspective of validity, Sandqvist’s semantics can also be viewed as the natural disjunction in a category of sheaves. (shrink)

Marc Lange argues that proofs by mathematical induction are generally not explanatory because inductive explanation is irreparably circular. He supports this circularity claim by presenting two putative inductive explanantia that are one another’s explananda. On pain of circularity, at most one of this pair may be a true explanation. But because there are no relevant differences between the two explanantia on offer, neither has the explanatory high ground. Thus, neither is an explanation. I argue that there is (...) no important asymmetry between the two cases because they are two presentations of the same explanation. The circularity argument requires a problematic notion of identity of proofs. I argue for a criterion of proof individuation that identifies the two proofs Lange offers. This criterion can be expressed in two equivalent ways: one uses the language of homotopy type theory, and the second assigns algebraic representatives to proofs. Though I will concentrate on one example, a criterion of proof identity has much broader consequences: any investigation into mathematical practice must make use of some proof-individuation principle. (shrink)

In this paper, we analyse conceptions of induction and certainty in Wolff and Crusius, highlighting their competing conceptions of physics. We discuss (i) the perspective of Wolff, who assigned induction an important role in physics, but argued that physics should be an axiomatic science containing certain statements, and (ii) the perspective of Crusius, who adopted parts of the ideal of axiomatic physics but criticized the scope of Wolff’s ideal of certain science. Against interpretations that take Wolff’s proofs in physics (...) to be based on empirical statements that are probable, we show how inductively established statements, coupled with assumptions concerning the uniformity of nature, can be certain according to Wolff. We examine Crusius’ little known work on physics and show that he attacked Wolff, arguing that many statements of physics are probable. We provide a discussion of the reception of methodological rules similar to Newton’s Regulae Philosophandi, discussing different interpretations of such rules. We conclude by briefly hinting at how the debate between Wolff and Crusius provides historical context for understanding Kant’s philosophy of natural science. (shrink)

This paper is an attempt at a vindication of induction. The point of departure is that induction requires a justification and that the only kind of justification possible is a vindication. However traditional vindications of induction have rested on unjustified assumptions about the aim of induction. This vindication takes the end pursued in induction simply to be correct prediction. It is argued that induction is the only reasonable way of pursuing this end because induction is the only objective method of (...) predicting. While objectivity is not a necessary condition for successful predicting, it is a sufficient condition for the possession of a property that is, in turn, a necessary condition for correct predicting; and further there could be no reason to expect any non-objective method to possess this necessary condition for success. Induction is the only objective method of predicting for it alone uses facts which must be identified by any predicting method based on rules; and it alone uses these facts according to a relationship that must hold between facts and any meaningful empirical statement. Any alternative to induction must assume that the factual regularities that have been detected will alter in the future and the way they are expected to alter must be arbitrary. It is further claimed that this vindication is proof against the criticisms that have been advanced against practicalist justifications of induction. (shrink)

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 Π1 0 and, similarly, for any class Π n 0 . 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 IΣ n , IΣ n − , IΠ n − 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 Σ1-reflection principle for T is Σ2-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. (shrink)

In this paper we calibrate the exact proof-theoretic strength of Kruskal's theorem, thereby giving, in some sense, the most elementary proof of Kruskal's theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction.

I focus on three preoccupations of recent writings on proof. -/- I. The role and possible effects of empirical reasoning in mathematics. Do recent developments (specifically, the computer-assisted proof of the 4CT) point to something essentially new as regards the need for and/or effects of using broadly empirical and inductive reasoning in mathematics? In particular, should we see such things as the computer-assisted proof of the 4CT as pointing to the existence of mathematical truths of which we cannot have (...) a priori knowledge? -/- 2. The role of formalization in proof. What are the patterns ofinference according to which mathematical reasoning naturally proceeds? Are they of 'local' character (i.e. sensitive to the subject-matter of the reasoning concerned) or 'global' character (i.e. invariant across all subject-matters)? Finally, what if any relationship is there (a) between the patterns of inference manifest in a proof and its explanatory capacity and (b) between explanatory capacity and rigor? -/- 3. Diagrams and their role in mathematical reasoning. What essentially is diagrammatic reasoning, and what is the nature and basis of its usefulness? Can it play a justificative role in the development of mathematical knowledge and, more particularly, in genuine proof? Finally, does the use of diagrammatic reasoning force an adjustment either in our conception of rigor or in our view of its importance? (shrink)

In intuitionistic generalized predicative systems as constructive set theory, or constructive type theory, two categories have been proposed to play the role of the category of locales: the category FSp of formal spaces, and its full subcategory FSpi of inductively generated formal spaces. Considered in impredicative systems as the intuitionistic set theory IZF, FSp and FSpi are both equivalent to the category of locales. However, in the mentioned predicative systems, FSp fails to be closed under basic constructions such as that (...) of finite products, while FSpi is complete , but does not contain some naturally occurring classes of formal spaces. Moreover, completeness of FSpi only holds under the assumption of strong principles for the existence of inductively defined sets.In this paper, working in the context of constructive set theory, the category ImLoc of imaginary locales is introduced. ImLoc is complete already over weak formulations of constructive set theory, contains FSp and FSpi as full subcategories, and, as FSp and FSpi, is equivalent to the category of locales in IZF.Applications of the concept of imaginary locale to a proof of a point-free version of Tychonoff embedding theorem, and to set-representation theorems, are then presented. (shrink)

Mathematicians do not claim to know a proposition unless they think they possess a proof of it. For all their confidence in the truth of a proposition with weighty non-deductive support, they maintain that, strictly speaking, the proposition remains unknown until such time as someone has proved it. This article challenges this conception of knowledge, which is quasi-universal within mathematics. We present four arguments to the effect that non-deductive evidence can yield knowledge of a mathematical proposition. We also show that (...) some of what mathematicians take to be deductive knowledge is in fact non-deductive. 1 Introduction2 Why It Might Matter3 Two Further Examples and Preliminaries4 An Exclusive Epistemic Virtue of Proof?5 Analyses of Knowledge6 The Inductive Basis of Deduction7 Physical to Mathematical Linkages8 Conclusion. (shrink)

This paper presents proof that Buss's S22 can prove the consistency of a fragment of Cook and Urquhart's PV from which induction has been removed but substitution has been retained. This result improves Beckmann's result, which proves the consistency of such a system without substitution in bounded arithmetic S12. Our proof relies on the notion of "computation" of the terms of PV. In our work, we first prove that, in the system under consideration, if an equation is proved and either (...) its left- or right-hand side is computed, then there is a corresponding computation for its right- or left-hand side, respectively. By carefully computing the bound of the size of the computation, the proof of this theorem inside a bounded arithmetic is obtained, from which the consistency of the system is readily proven. This result apparently implies the separation of bounded arithmetic because Buss and Ignjatović stated that it is not possible to prove the consistency of a fragment of PV without induction but with substitution in Buss's S12. However, their proof actually shows that it is not possible to prove the consistency of the system, which is obtained by the addition of propositional logic and other axioms to a system such as ours. On the other hand, the system that we have considered is strictly equational, which is a property on which our proof relies. (shrink)

In this paper1, we explore some positive elements from the Epicurean position on mathematics. Is induction important in mathematical practice or useful in proof? Does atomism appear in mathematics and in what ways? Keywords: Epicurus, induction, Polya, proof, atomism.

This two-volume work, first published in 1843, was John Stuart Mill's first major book. It reinvented the modern study of logic and laid the foundations for his later work in the areas of political economy, women's rights and representative government. In clear, systematic prose, Mill (1806–73) disentangles syllogistic logic from its origins in Aristotle and scholasticism and grounds it instead in processes of inductive reasoning. An important attempt at integrating empiricism within a more general theory of human knowledge, the (...) work constitutes essential reading for anyone seeking a full understanding of Mill's thought. Volume 1 contains Mill's introduction, which elaborates upon his definition of logic as 'not the science of Belief, but the science of Proof, or Evidence'. It also features discussions of the central components of logical reasoning - propositions and syllogisms - in relation to Mill's theories of inductive reasoning and experimental method. (shrink)

In this thesis, we study the least fixed point principle in a constructive setting. A constructive theory of functions and sets has been developed by Feferman. This theory deals both with sets and with functions over sets as independent notions. In the language of Feferman's theory, we are able to formulate the least fixed point principle for monotone inductive definitions as: every operation on classes to classes which satisfies the monotonicity condition has a least fixed point. This is called (...) the principle of monotone inductive definition. Furthermore, we may formulate this principle in a uniform way as: there is an operation which maps a monotone operation to its least fixed point. This is called the principle of uniform monotone inductive definition. Feferman raised the question of the strength of the principle of monotone inductive definition when adjoined to his theory . This question is our primary concern in this thesis. ;Our main results are the consistency of both the principle of monotone inductive definition and the principle of uniform monotone inductive definition adjoined to his theory, and the proof theoretical equivalence between the principle of monotone inductive definition with Feferman's system without the inductive generation axiom and the system of the Pi-one-one Comprehension Axiom. Determination of the proof-theoretical strength of the principle of monotone inductive definition adjoined to Feferman's theory still remains open. ;The consistency of both the principle of monotone inductive definition and the principle of uniform monotone inductive definition with Feferman's theory will be achieved by constructing their models in set theoretical sense. The proof theoretical equivalence between the principle of monotone inductive definition with Feferman's system without the inductive generation axiom and the system of the Pi-one-one Comprehension Axiom can be obtained by a careful examination of the model construction for the principle of monotone inductive definition with Feferman's system without the inductive generation axiom, which is parallel to the model construction for the principle of monotone inductive definition with his theory. (shrink)

Induction–recursion is a powerful definition method in intuitionistic type theory. It extends inductive definitions and allows us to define all standard sets of Martin-Löf type theory as well as a large collection of commonly occurring inductive data structures. It also includes a variety of universes which are constructive analogues of inaccessibles and other large cardinals below the first Mahlo cardinal. In this article we give a new compact formalization of inductive–recursive definitions by modeling them as initial algebras (...) in slice categories. We give generic formation, introduction, elimination, and equality rules generalizing the usual rules of type theory. Moreover, we prove that the elimination and equality rules are equivalent to the principle of the existence of initial algebras for certain endofunctors. We also show the equivalence of the current formulation with the formulation of induction–recursion as a reflection principle given in Dybjer and Setzer 93). Finally, we discuss two type-theoretic analogues of Mahlo cardinals in set theory: an external Mahlo universe which is defined by induction–recursion and captured by our formalization, and an internal Mahlo universe, which goes beyond induction–recursion. We show that the external Mahlo universe, and therefore also the theory of inductive–recursive definitions, have proof-theoretical strength of at least Rathjen's theory KPM. (shrink)

An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic, natural deduction and the normalization theorems, the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications (...) of these results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy of mathematics. (shrink)

Propositional logic -- Predicate logic -- Proof strategies and diagrams -- Mathematical induction -- Set theory -- Functions -- Relations -- Core concepts in abstract algebra -- Core concepts in real analysis.