Lectures in Logic and Set Theory

arXiv: History and Overview, 2014

My purpose is to examine some concepts of mathematical logic, which have been studied by Carlo Cellucci. Today the aim of classical mathematical logic is not to guarantee the certainty of mathematics, but I will argue that logic can help us to explain mathematical activity; the point is to discuss what and in which sense logic can "explain". For example, let's consider the basic concept of an axiomatic system: an axiomatic system can be very useful to organize, to present, and to clarify mathematical knowledge. And, more importantly, logic is a science with its own results: so, axiomatic systems are interesting also because we know several revealing theorems about them. Similarly, I will discuss other topics such as mathematical definitions, and some relationships between mathematical logic and computer science. I will also consider these subjects from an educational point of view: can logical concepts be useful in teaching and learning elementary mathematics?

haverford.edu

Catalyzed by the failure of Frege's logicist program, logic and the foundations of mathematics first became a philosophical topic in Europe in the early years of the twentieth century. Frege had aimed to show that logic constitutes the foundation for mathematics in the sense of providing both the primitive concepts in terms of which mathematical concepts were to be defined and the primitive truths on the basis of which mathematical truths were to be proved. 1 Russell's paradox showed that the project could not be completed, at least as envisaged by Frege. It nevertheless seemed clear to many that mathematics must be founded on something, and over the first few decades of the twentieth century four proposals emerged: two species of logicism, namely ramified type theory as developed in Russell and Whitehead's Principia and Zermelo-Frankel set theory, Hilbert's finitist program (a species of formalism), and finally Brouwerian intuitionism. 2 Across the Atlantic, already by the time Russell had discovered his famous paradox, the great American pragmatist Charles Sanders Peirce was developing a radically new non-foundationalist picture of mathematics, one that, through the later influence of Quine, Putnam, and Benacerraf, would profoundly shape the course of the philosophy of mathematics in the United States.

This paper brings up some important points about logic, e.g., mathematical logic, and also an inconsistence in logic as per Gödel's incompleteness theorems which state that there are mathematical truths that are not decidable or provable. These incompleteness theorems have shaken the solid foundation of mathematics where innumerable proofs and theorems have a place of pride. The great mathematician David Hilbert had been much disturbed by them. There are much long unsolved famous conjectures in mathematics, e.g., the twin primes conjecture, the Goldbach conjecture, the Riemann hypothesis, etc. Perhaps, by Gödel's incompleteness theorems the proofs for these famous conjectures will not be possible and the numerous mathematicians attempting to find the solutions for these conjectures are simply banging their heads against the metaphorical wall. Besides mathematics, Gödel's incompleteness theorems will have ramifications in other areas involving logic. This paper looks at the ramifications of the incompleteness theorems, which pose the serious problem of inconsistency, and offers a solution to this dilemma. The paper also looks into the apparent inconsistence of the axiomatic method in mathematics. [Published in international mathematics journal. Acknowledgments: The author expresses his gratitude to the referees and the Editor-in-Chief for their valuable comments in strengthening the contents of this paper.]

Springer eBooks, 2021

Logic, Epistemology, and the Unity of Science aims to reconsider the question of the unity of science in light of recent developments in logic. At present, no single logical, semantical or methodological framework dominates the philosophy of science. However, the editors of this series believe that formal frameworks, for example, constructive type theory, deontic logics, dialogical logics, epistemic logics, modal logics, and proof-theoretical semantics, have the potential to cast new light on basic issues in the discussion of the unity of science. This series provides a venue where philosophers and logicians can apply specific systematic and historic insights to fundamental philosophical problems. While the series is open to a wide variety of perspectives, including the study and analysis of argumentation and the critical discussion of the relationship between logic and philosophy of science, the aim is to provide an integrated picture of the scientific enterprise in all its diversity. This book series is indexed in SCOPUS.

Epistemology versus Ontology, 2012

The question, whether second order logic is a better foundation for mathematics than set theory, is addressed. The main difference between second order logic and set theory is that set theory builds up a transfinite cumulative hierarchy while second order logic stays within one application of the power sets. It is argued that in many ways this difference is illusory. More importantly, it is argued that the often stated difference, that second order logic has categorical characterizations of relevant mathematical structures, while set theory has non-standard models, amounts to no difference at all. Second order logic and set theory permit quite similar categoricity results on one hand, and similar non-standard models on the other hand.

In the first part of this investigation we highlight two, seemingly irreconcilable, beliefs that suggest an impending crisis in the teaching, research, and practice of mathematics: (1) the belief, with increasing, essentially faith-based, conviction and authority amongst academics that a first-order Set Theory such as ZFC can be treated as the lingua franca of mathematics, since its theorems-even when unfalsifiable-can be treated as 'knowledge' because they are finite proof sequences which are entailed finitarily by self-evidently Justified True Beliefs; and (2) the gradually emerging, but powerfully rooted in economic imperatives, belief that only those Justified True Beliefs can be treated as 'knowledge' which are, further, categorically communicable as Factually Grounded Beliefs-hence as comprehensible pre-formal 'mathematical truths'-by any intelligence that lays claims to a mathematical lingua franca. We argue that this seeming irreconcilability merely reflects a widening chasm between an increasing underestimation of the value to society of 'semantic arithmetical truth', and an increasing over-estimation of the value to society of 'syntactic set-theoretical provability'; a chasm which must be narrowed and bridged explicitly to avoid lending an illusory legitimacy-by-omission to the perilous concept of 'alternative facts'. We thus proffer a Complementarity Thesis which seeks to recognize that mathematical 'provability' and mathematical 'truth' need to be interdependent and complementary, 'evidence-based', assignments-by-convention to the formulas of a formal mathematical language; where (a) the goal of mathematical 'proofs' may be viewed as seeking to ensure that any mathematical language intended to formally represent our pre-formal conceptual metaphors and their inter-relatedness is unambiguous, and free from contradiction; whilst (b) the goal of mathematical 'truths' must be viewed as seeking to 'validate' that any such representation does, indeed, uniquely identify and adequately represent such metaphors and their inter-relatedness. Our thesis is that, by universally ignoring the need to introduce goal (b) in our curriculum, the teaching of, and research in, mathematics at the postgraduate level cannnot justify its value to society beyond the mere intellectual stimulation of individual minds. In the second part we appeal to some recent-and hitherto unsuspected-unarguably constructive Tarskian interpretations, of the first-order Peano Arithmetic PA, which establish PA as both finitarily consistent, and categorical. Since we also show that the second order subsystem ACA 0 of Peano Arithmetic and PA cannot both be assumed provably consistent, we conclude that there can be no mathematical, or meta-mathematical, proof of consistency for Set Theory. Hence the theorems of any Set Theory are not sufficient for distinguishing between (i) what we can believe to be a 'mathematical truth'; (ii) what we can evidence as a 'mathematical truth'; and (iii) what we ought not to believe is a 'mathematical truth'; whilst the theorems of the first-order Peano Arithmetic PA are sufficient for distinguishing between (i), (ii) and (iii). We conclude from this that the holy grail of mathematics ought to be 'arithmetical truth', not 'set-theoretical proof'.

The British Journal for the Philosophy of Science, 2003

In The Foundations of Mathematics in the Theory of Sets, John Mayberry attacks the 2000-year-old problem of accounting for the foundations of mathematics. His account comes in three interrelated parts: determining exactly what one should (and should not) expect from a foundation; arguing that set theory can in fact provide such a foundation, and presenting a novel version of set theory (or at least a novel exposition of traditional set theory) which can fulfil this foundational role. He addresses the first of these issues (in Chapters 1 and 6) by arguing that the most important aspect of mathematics is the axiomatic method, and thus a foundation for mathematics must provide a justification of this method (but need provide little more). Axiom systems, according to Mayberry, isolate particular mathematical structures (or are meant to), and all that is left for a foundation to supply is a guarantee that some appropriate structure exists that satisfies the axioms: For when we employ the axiomatic method, the only special subject matter that we need acknowledge as belonging to mathematics is that of sets [. . .]. All that was required, historically, to replace the traditional methods completely was to lay down three axiomatic theories [analysis, arithmetic, and geometry] [. . .] and then to show how [these theories] could be reconstructed by straightforward algebraic and set-theoretical methods applied to models of those theories. (p. 204) He quite correctly points out that the formal axiomatization of Zermelo-Fraenkel set theory is an inappropriate place to look for such a foundation, since this is, like analysis, a mathematical theory in need of a foundation, not a candidate for providing one. He argues, therefore, that what is needed is an