Against Set Theory

2013

According to Cantor (Mathematische Annalen 21:545-586, 1883; Cantor's letter to Dedekind, 1899) a set is any multitude which can be thought of as one ("jedes Viele, welches sich als Eines denken läßt") without contradiction-a consistent multitude. Other multitudes are inconsistent or paradoxical. Set theoretical paradoxes have common root-lack of understanding why some multitudes are not sets. Why some multitudes of objects of thought cannot themselves be objects of thought? Moreover, it is a logical truth that such multitudes do exist. However we do not understand this logical truth so well as we understand, for example, the logical truth ∀x x = x. In this paper we formulate a logical truth which we call the productivity principle. Rusell (Proc Lond Math Soc 4(2):29-53, 1906) was the first one to formulate this principle, but in a restricted form and with a different purpose. The principle explicates a logical mechanism that lies behind paradoxical multitudes, and is understandable as well as any simple logical truth. However, it does not explain the concept of set. It only sets logical bounds of the concept within the framework of the classical two valued ∈-language. The principle behaves as a logical regulator of any theory we formulate to explain and describe sets. It provides tools to identify paradoxical classes inside the theory. We show how the known paradoxical classes follow from the productivity principle and how the principle gives us a uniform way to generate new paradoxical classes. In the case of ZFC set theory the productivity principle shows that the limitation of size principles are of a restrictive nature and that they do not explain which classes are sets. The productivity principle, as a logical regulator, can have a definite heuristic role in the development of a consistent set theory. We sketch such a theorythe cumulative cardinal theory of sets. The theory is based on the idea of cardinality

1999

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.

2014

Bulletin of Symbolic Logic, 2007

Kurt Gödel (1906–1978) with his work on the constructible universe L established the relative consistency of the Axiom of Choice (AC) and the Continuum Hypothesis (CH). More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic constructions and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel's work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently p...

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'.

Notices of the American Mathematical Society, 2006

Set Theory is a formalization of the existence and fundamental properties of mathematical objects as collections of elements and/or elements included in collections. Its formulation is so basic and comprehensive that it has been postulated as the foundation of all mathematics. Perhaps, the major achievement of Set Theory is that, after being criticized by many reputable mathematicians and philosophers since its appearance, it is now commonly accepted as the primary explanation of the most basic components of mathematics: numbers; and not only the numbers we have needed or we may ever need but all the numbers that could potentially exist. In Set Theory, an infinite sequence of numbers exists not as the mere projection of a construction algorithm but as a complete and self-identical mathematical object: a set. In Set Theory the words infinite and infinity do not refer to the property of growing endlessly (potential infinity) but to a definite magnitude; a number; the actual infinity. As a result of such a conception, Set Theory arrives at the conclusion that there exist infinitely many infinities, each one with a different value. The set of postulates, proofs and theorems used to justify the existence of such infinities is commonly known as Transfinite Set-Theory. The first part of this work shows how some of the properties and theorems applied to infinite sets, in Set Theory, necessarily lead to internal and fundamental contradictions under classical logic, even when the idea of actual infinity is accepted. Throughout the second part, motivated by the necessity of an alternative to Transfinite Set-Theory, due to the incapacity of such a theory to explain some of the findings shown in the first part (especially the proof of the existence of as many rational as irrational numbers), the author develops a theory to provide a better understanding of infinite sequences of numbers.

INTENTIO, 2024

The intended scope of this article is to review the Continuum Hypothesis, CH, in foundational mathematics from the viewpoint of a phenomenologically influenced philosopher. Given the capital importance of Cantor’s 19th century conjecture about the cardinality of continuum and the relevance it has acquired over the years in matters of mathematical ontology it is natural to motivate a discussion well beyond its place in foundational mathematics proper. This means that except for the continuing research toward its resolution by new and powerful theories following Cohen’s pioneering forcing method in the 60s, one has come to inquire into the metatheoretical or even extra-theoretical nature of the Continuum Hypothesis, something that implies a philosophical inquiry into the ontological status of the question as such and into the ways it may be shaped by our natural intuitions of the continuous and the discrete. On the latter prompt, among others, and discarding platonistic tendencies I set out to provide an interpretation along subjective-constitutional tracks of the ontological status of the CH and ultimately of the ways it may be reducible to a question of the constitution of continuous unity in subjective-temporal terms. In the more formal part, I have tried to argue against a certain operationally motivated attempt at refuting CH as well as against attempts to resolve the issue of its undecidability relying on the more advanced large cardinals plus the inner models theory.