Some observations on the logical foundations of inductive theorem proving

In this paper we study the logical foundations of automated inductive theorem proving. To that aim we first develop a theoretical model that is centered around the difficulty of finding induction axioms which are sufficient for proving a goal. Based on this model, we then analyze the following aspects: the choice of a proof shape, the choice of an induction rule and the language of the induction formula. In particular, using model-theoretic techniques, we clarify the relationship between notions of inductiveness that have been considered in the literature on automated inductive theorem proving. This is a corrected version of the paper arXiv:1704.01930v5 published originally on Nov.~16, 2017

Connecting the provable with the unprovable: phase transitions for unprovability

Why are some theorems not provable in certain theories of mathematics? Why are most theorems from existing mathematics provable in very weak systems? Unprovability theory seeks answers for those questions. Logicians have obtained unprovable statements which resemble provable statements. These statements often contain some condition which seems to cause unprovability, as this condition can be modified, using a function parameter, in such a manner as to make the theorem provable. It turns out that in many cases there is a phase transition: By modifying the parameter slightly one changes the theorem from provable to unprovable. We study these transitions with the goal of gaining more insights into unprovability

The Consistency of Arithmetic

This paper offers an elementary proof that formal arithmetic is consistent. The system that will be proved consistent is a first-order theory R♯, based as usual on the Peano postulates and the recursion equations for + and ×. However, the reasoning will apply to any axiomatizable extension of R♯ got by adding classical arithmetical truths. Moreover, it will continue to apply through a large range of variation of the un- derlying logic of R♯, while on a simple and straightforward translation, the classical first-order theory P♯ of Peano arithmetic turns out to be an exact subsystem of R♯. Since the reasoning is elementary, it is formalizable within R♯ itself; i.e., we can actually demonstrate within R♯ (or within P♯, if we care) a statement that, in a natural fashion, asserts the consistency of R♯ itself. The reader is unlikely to have missed the significance of the remarks just made. In plain English, this paper repeals Goedel’s famous second theorem. (That’s the one that asserts that sufficiently strong systems are inadequate to demonstrate their own consistency.) That theorem (or at least the significance usually claimed for it) was a mis- take—a subtle and understandable mistake, perhaps, but a mistake nonetheless. Accordingly, this paper reinstates the formal program which is often taken to have been blasted away by Goedel’s theorems— namely, the Hilbert program of demonstrating, by methods that everybody can recognize as effective and finitary, that intuitive mathematics is reliable. Indeed, the present consistency proof for arithmetic will be recognized as correct by anyone who can count to 3. (So much, indeed, for the claim that the reliability of arithmetic rests on transfinite induction up to ε0, and for the incredible mythology that underlies it.

Provably total recursive functions and MRDP theorem in Basic Arithmetic and its extensions

We study Basic Arithmetic, BA introduced by W. Ruitenburg. BA is an arithmetical theory based on basic logic which is weaker than intuitionistic logic. We show that the class of the provably recursive functions of BA is a proper sub-class of primitive recursive functions. Three extensions of BA, called BA+U, BA_c and EBA are investigated with relation to their provably recursive functions. It is shown that the provably recursive functions of these three extensions of BA are exactly primitive recursive functions. Moreover, among other things, it is shown that the well-known MRDP theorem doesn't hold in BA, BA+U, BA_c, but holds in EBA

Abstraction Logic: A New Foundation for (Computer) Mathematics

Abstraction logic is a new logic, serving as a foundation of mathematics. It combines features of both predicate logic and higher-order logic: abstraction logic can be viewed both as higher-order logic minus static types as well as predicate logic plus operators and variable binding. We argue that abstraction logic is the best foundational logic possible because it maximises both simplicity and practical expressivity. This argument is supported by the observation that abstraction logic has simpler terms and a simpler notion of proof than all other general logics. At the same time, abstraction logic can formalise both intuitionistic and classical abstraction logic, and is sound and complete for these logics and all other logics extending deduction logic with equality

Categoricity

After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those involving the distinction between characterizing a system and axiomatizing the truths of a syste

Unprovability and phase transitions in Ramsey theory

The first mathematically interesting, first-order arithmetical example of incompleteness was given in the late seventies and is know as the Paris-Harrington principle. It is a strengthened form of the finite Ramsey theorem which can not be proved, nor refuted in Peano Arithmetic. In this dissertation we investigate several other unprovable statements of Ramseyan nature and determine the threshold functions for the related phase transitions. Chapter 1 sketches out the historical development of unprovability and phase transitions, and offers a little information on Ramsey theory. In addition, it introduces the necessary mathematical background by giving definitions and some useful lemmas. Chapter 2 deals with the pigeonhole principle, presumably the most well-known, finite instance of the Ramsey theorem. Although straightforward in itself, the principle gives rise to unprovable statements. We investigate the related phase transitions and determine the threshold functions. Chapter 3 explores a phase transition related to the so-called infinite subsequence principle, which is another instance of Ramsey’s theorem. Chapter 4 considers the Ramsey theorem without restrictions on the dimensions and colours. First, generalisations of results on partitioning α-large sets are proved, as they are needed later. Second, we show that an iteration of a finite version of the Ramsey theorem leads to unprovability. Chapter 5 investigates the template “thin implies Ramsey”, of which one of the theorems of Nash-Williams is an example. After proving a more universal instance, we study the strength of the original Nash-Williams theorem. We conclude this chapter by presenting an unprovable statement related to Schreier families. Chapter 6 is intended as a vast introduction to the Atlas of prefixed polynomial equations. We begin with the necessary definitions, present some specific members of the Atlas, discuss several issues and give technical details

Numerical Methods for Infinite Decision-Making Processes

The new computational methodology due to Yaroslav Sergeyev (see [25–27]) makes it possible to evaluate numerically the terminal features of complete, sequential decision-making processes. By standard numerical methods, these processes have indeterminate features or seem to support paradoxical conclusions. We show that they are better regarded as a class of problems for which the numerical methods based on Sergeyev’s methodology provide a uniform technique of resolution