Set Theory as a Foundation

Set theory is a rich and beautiful subject whose fundamental concepts permeate virtually every branch of mathematics. One could say that set theory is a unifying theory for mathematics, since nearly all mathematical concepts and results can be formalized within set theory. This textbook is meant for an upper undergraduate course in set theory. In this text, the fundamentals of abstract sets, including relations, functions, the natural numbers, order, cardinality, transfinite recursion, the axiom of choice, ordinal numbers, and cardinal numbers, are developed within the framework of axiomatic set theory. The reader will need to be comfortable reading and writing mathematical proofs. The proofs in this textbook are rigorous, clear, and complete, while remaining accessible to undergraduates who are new to upper-level mathematics. Exercises are included at the end of each section in a chapter, with useful suggestions for the more challenging exercises.

The aim of this tutorial is to present a brief overview of the theory of rough sets from the perspective of its mathematical foundations, history of development as well as connections with other branches of mathematics and informatics. The content concerns both the theoretical and practical aspects of applications. The above mentioned target of the tutorial will be covered in two parts. In the first part we would aim to present the introduction to rough sets and the second part will focus on the connections with other branches of mathematics and informatics.

A survey was conducted on the frequency of usage of psychologists' names in early introductory psychology textbooks (1920–1939) and recent introductory psychology textbooks (1968–1971) in order to assess differences between these periods. Using eponymy indexes, such as naming ratios which were derived from frequency of naming divided by textbook pages, a methodology was developed for comparing early and recent textbooks. Results showed that: (1) the average naming ratio for early textbooks was significantly lower than the ratio for recent textbooks; (2) there were significant differences in naming between early and recent textbooks in terms of various chapter headings and subject materials; and (3) there were significant differences in naming particular psychologists, e.g., Freud and Pavlov, with greater frequency occurring in the recent over the early textbooks. Results were discussed within the context of E. G. Boring's writings concerning eponymy and the future state of psychology.

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.

Axiomatic set theory is almost universally accepted as the basic theory which provides the foundations of mathematics, and in which the whole of present day mathematics can be developed. As such, it is the most natural framework for Mathematical Knowledge Management. However, in order to be used for this task it is necessary to overcome serious gaps that exist between the "official" formulations of set theory (as given e.g. by formal set theory ZF) and actual mathematical practice. In this work we present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity.

Homotopy Type Theory (HoTT) is a putative new foundation for mathematics grounded in constructive intensional type theory, that offers an alternative to the foundations provided by ZFC set theory and category theory. This paper explains and motivates an account of how to define, justify and think about HoTT in a way that is self-contained, and argues that it can serve as an autonomous foundation for mathematics. We first consider various questions that a foundation for mathematics might be expected to answer, and find that the standard formulation of HoTT as presented in the ``HoTT Book'' does not answer many of them. More importantly, the way HoTT is developed in the HoTT Book suggests that it is not a candidate \emph{autonomous} foundation since it explicitly depends upon other fields of mathematics, in particular homotopy theory. We give an alternative presentation of HoTT that does not depend upon sophisticated ideas from other parts of mathematics, and in particular makes no reference to homotopy theory (but is compatible with the homotopy interpretation). Our elaboration of HoTT is based on a new interpretation of types as \emph{mathematical concepts}, which accords with the intensional nature of the type theory.

From the Publisher: Introduces the process of abstraction, studies two elementary structures on sets, and covers the necessary generalities concerning algebraic structures. Presents powerful abstract mathematical concepts from algebra and combinatorics, supported by concrete applications. All background material is provided, including elements of logic, set theory, abstract algebra, linear algebra, and graph theory. Each chapter develops a new mathematical concept, then shows how to apply it. Includes numerous end-of-chapter problems and exercises.

BOOK REVIEWS 361 one hand, or realists, on the other. Dewey placed knowing within the wider view of experience ; but realists interpreted the term experience to be what the idealists called consciousness, and thus Dewey was to them a disguised idealist or at best "a half-hearted naturalist," as Santayana called him. To some idealists, Dewey was a confused realist, to others he was a materialist , and to others he was a Cartesian dualist. Dewey's account of the knowing process that developed out of the methods of experimental science had difficulties in making its way into the minds of philosophers already committed to realism or idealism. Dicker's attempt to relate Dewey to the different contexts of epistemological writing is admirable but a formidable task, a task that requires more space and more detail. Be that as it may, this reviewer will direct beginning students to Dicker's monograph as a starting place in understanding this difficult theory. S. MORalS EAMES Southern Illinois University, Carbondale Cora Diamond, ed. Wittgenstein's Lectures on the Foundations of Mathematics: Cambridge, 1939. From the notes of R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies. Ithaca: Corneil University Press, 1976. Pp. 300. $18.50. From 1929 through 1932 and again from 1937 through 1944 Ludwig Wittgenstein wrote extensively on the philosophy of mathematics. Manuscripts from the earlier period are close in concept to his Tractatus Logico-Philosophicus. Those of the later period, which are conceptually close to his Philosophical Investigations, have been published as Remarks on the Foundations of Mathematics (1956). The Remarks, as it turns out, do not exhaust the available literature by Wittgenstein devoted exclusively to the philosophy of mathematics. In 1939 he delivered thirty-one lectures on the foundations of mathematics at Cambridge University. These lectures, which provide extremely rich insights into Wittgenstein's thought, make up the present volume. Wittgenstein lectured without notes. In reconstructing his lectures it has been possible to draw on student notes by R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies. The result is surprisingly coherent. Where student accounts stand in contradiction or are fragmentary the editor provides cautionary footnotes. References are also provided signaling closely related passages in the Remarks and Investigations. There is little in these lectures to surprise those familiar with Wittgenstein's later philosophy . The enemy here, as elsewhere, is the Iogicism of Russell and Frege, with its close ties to Platonism and Platonic interpretations of Cantorian arithmetic, and its promise of an ideal language. Hilbert's formalism is mentioned only in passing, while the intuitionism of Brouwer and others is dismissed as sheer "bosh"--a move that relieves the lecturer of having to consider how near his own position is to intuitionism. Mathematics, Wittgenstein argues, is an activity, not a series of discoveries of an increasingly known realm of mathematical "objects." The temptation to suppose that when we are dealing with Cantorian infinite sets we are confronted with something immense must be thwarted by the realization that in mathematics we are dealing with "rules of expression." The child who has learned ~ o multiplications is not confronted with something unimaginably gigantic; he has simply learned a rule for proceeding in mathematics. Mathematical concepts must be viewed as human linguistic inventions. Exchanges between Wittgenstein and his students are included by the editor. Many of these (especially the discussions with A. M. Turing, John Wisdom, and Casimir Lewy) are extremely helpful. For example: [To Turing] Before we stop, could you say whether you really think that it is the contradiction which gets you into trouble-the contradiction in logic? Or do you see that it is something quite different?-I don't say that a contradiction may not get you into trouble. Of course it may. 362 HISTORY OF PHILOSOPHY Turing: 1think that with the ordinary kind of rules which one uses in logic, if one can get into contradictions , then one can get into trouble. Wittgenstein: But does this mean that withcontradictions one musI get into trouble? Or do you mean that the contradiction may tempt one into trouble? As a matter of fact it doesn't. No one has ever yet got into trouble from a contradiction in...

Metapredicativity is a new general term in proof theory which describes the analysis and study of formal systems whose proof-theoretic strength is beyond the Feferman-Schutte ordinal Γ0 but which are nevertheless amenable to purely predicative methods. Typical examples of formal systems which are apt for scaling the initial part of metapredicativity are the transfinitely iterated fixed point theories IDα whose detailed proof-theoretic analysis is given by Jager, Kahle, Setzer and Strahm in [18]. In this paper we assume familiarity with [18]. For natural extensions of Friedman’s ATR that can be measured against transfinitely iterated fixed point theories the reader is referred to Jager and Strahm [20]. In the mid seventies, Feferman [3, 4] introduced systems of explicit mathematics in order to provide an alternative foundation of constructive mathematics. More precisely, it was the origin of Feferman’s program to give a logical account of Bishop-style constructive mathematics. Right from the beginning, systems of explicit mathematics turned out to be of general interest for proof theory, mainly in connection with the proof-theoretic analysis of subsystems of first and second order arithmetic and set theory, cf. e.g. Jager [15] and Jager and Pohlers [19]. More recently, systems of explicit mathematics have been used to develop a general logical framework for functional programming and type theory, where it is possible to derive correctness and termination properties of functional programs. Important references in this connection are Feferman [6, 7, 9] and Jager [17]. Universes are a frequently studied concept in constructive mathematics at least since the work of Martin-Lof, cf. e.g. Martin-Lof [23] or Palmgren [27] for

Set theory, logic and category theory lie at the foundations of mathematics, and have a dramatic effect on the mathematics that we do, through the Axiom of Choice, Godel's Theorem, and the Skolem Paradox. But they are also rich mathematical theories in their own right, contributing techniques and results to working mathematicians such as the Compactness Theorem and module categories. The book is aimed at those who know some mathematics and want to know more about its building blocks. Set theory is first treated naively an axiomatic treatment is given after the basics of first-order logic have been introduced. The discussion is su pported by a wide range of exercises. The final chapter touches on philosophical issues. The book is supported by a World Wibe Web site containing a variety of supplementary material.

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsSolomon FefermanSOLOMON FEFERMAN is professor of Mathematics and Philosophy, and Suppes Family Professor in Humanities and Sciences at Stanford University. He is past President of the Association for Symbolic Logic and is a Fellow of the American Academy of Arts and Sciences. He received his Ph.D. at U.C. Berkeley in 1957 under the direction of Alfred Tarski. He is Editor-in-Chief of the Collected Works of Kurt Gödel, and the author of numerous articles on mathematical logic, the foundations of mathematics, and the history of modern logic.

The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, i.e. $\textsf{ZFC}$ set theory, all mathematical objects are represented by sets, while ordinary, i.e. non-set theoretic, mathematics is represented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continuous functions on the reals and Baire space. We show that the logical strength of basic theorems named after Tietze, Heine, and Weierstrass, changes significantly upon the replacement of 'second-order representations' to 'third-order functions'. We discuss the implications and connections to the Reverse Mathematics program and its foundational claims regarding predicativist mathematics and Hilbert's program for the foundations of mathematics. Finally, we identify the problem caused by representations of continuous functions and formulate a criterion to avoid problematic codings within the bigger picture of representations.

The foundations of mathematics consist of two basic theories, the logical calculus and set theory, together with methodological research concerning these and other mathematical theories. The logical and the set theoretical concepts arise from an analysis of the subject matter of mathematics as well as from an analysis of the construction of mathematical concepts. Let us first consider the subject matter of mathematics.

In his later works, the great logician and mathematician Kurt Gödel concentrates his focus on the philosophical problems such as the implications of set theory, the grammar and philosophy of language, objectivity and relativity, the ontological proof of God’s existence, and phenomenology as an exact method. This essay explores how Gödel reads the philosophy (of logic and mathematics) of his time and why he turns his attention to Husserl’s phenomenology for describing the foundations of mathematics. To begin with, Gödel employs Husserl’s significant distinction between Weltanschauung (worldview) philosophy and philosophy as rigorous science: According to the Weltanschauung philosophy, the spirit of time constantly changes so that the ideas discussed and goals attempted are meant to be temporal, and not for the sake of eternal truths, but for that of their own perfection; philosophy as rigorous science, on the other hand, is supratemporal so that its aim is to discover absolute and timeless values. As for the worldview of his time, Gödel sees the development of philosophy and mathematics leaned toward skepticism, pessimism, and positivism. The antinomies of set theory, for instance shaked the grounds on which mathematics and logic are founded. Gödel, too, uses these paradoxes in his incompleteness theorems in order to prove that there are some statements which can neither be proved nor disproved within a system. That also means that arithmetic is not eligible to prove its own consistency. From this, however, Gödel does not come to a conclusion for a nihilism in mathematics and logic: These mere antinomies of set theory do not “necessarily” lead us to logical positivism, and neither to such a materialism, nor to any kind of pessimistic theory of knowledge. The incompleteness theorems assert that there are arithmetical propositions that are true but neither provable nor unprovable within its own calculus, so that arithmetic is intrinsically incomplete. However, instead of Alfred Tarski’s pathological view of examining the detections within the faulty system and then reforming the system all together, Gödel holds that we need to change our methods to find new patterns that describe the antinomies pointing to the unrecoverable reality of the mathematical world. Thus, Gödel does not follow any variation of the Weltanschauung philosophy of his time, either attempting to reduce mathematical realities to mathematical proofs in order to get rid of antinomies, or endeavoring to rescue a complete system of truths by a closed formal system, both Weltanschauung philosophies fail to set forth a realistic method. In this context, Gödel finds the task of phenomenology analogous to what he pursues in terms of a systematic framework for the foundations of mathematics. Husserl’s phenomenology, in Gödel’s account, proliferates the intuition of (mathematical) essences and provides a clarification of meaning of undefinable concepts, such as the antinomies of set theory. Applying the phenomenological reduction to the objective reality of the mathematical world, Gödel believes one obtains a clear experiential reality of the essential characteristics of (mathematical and logical) concepts. Briefly put, what Gödel finds in Husserl’s phenomenology that corresponds to his way of mathematical realism is a thoroughly designated method giving us mathematical essences back again.

If there is one phrase today that is prominent in the literature and conversation of all persons, professional and lay, and yet very nebulous, ambiguous, and meaningless in the minds of most of these persons, it is “modern math” or “the new math.” Now Modern Mathematics, as exemplified in such topics as Modern Algebra, Linear Algebra, Point-Set Topology, Algebraic Topology, Finite Mathematical Systems, Set Theory, and the like, is an exceedingly abstract, logical, axiomatic, well-structured system of knowledge. There is no nonsense in these branches of mathematics. They are pursued by students majoring in mathematics and science at the upper university and graduate levels of study. They have no place—no place—in elementary school mathematics. To do any of this mathematics in the elementary school would be more than nonsense—it would be insanity.