A Benacerraf problem for higher‐order metaphysics

On the Consistency Problem for Set Theory: An Essay on the Cantorian Foundations of Classical Mathematics (I)

Immanuel Kant’s Critique of Pure Reason is widely taken to be the starting point of the modern period of mathematics while David Hilbert was the last great mainstream mathematician to pursue important nineteenth century ideas. This two-volume work provides an overview of this important era of mathematical research through a carefully chosen selection of articles. They provide an insight into the foundations of each of the main branches of mathematics-algebra, geometry, number theory, analysis, logic and set theory--with narratives to show how they are linked. Classic works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare are reproduced in reliable translations and many selections from writers such as Gauss, Cantor, Kronecker and Zermelo are here translated for the first time. The collection is an invaluable source for anyone wishing to gain an understanding of the foundation of modern mathematics.

Immanuel Kant's Critique of Pure Reason is widely taken to be the starting point of the modern period of mathematics while David Hilbert was the last great mainstream mathematician to pursue important nineteenth cnetury ideas. This two-volume work provides an overview of this important era of mathematical research through a carefully chosen selection of articles. They provide an insight into the foundations of each of the main branches of mathematics--algebra, geometry, number theory, analysis, logic and set theory--with narratives to show how they are linked. Classic works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare are reproduced in reliable translations and many selections from writers such as Gauss, Cantor, Kronecker and Zermelo are here translated for the first time. The collection is an invaluable source for anyone wishing to gain an understanding of the foundation of modern mathematics.

Book Information The Foundations of Mathematics in the Theory of Sets. The Foundations of Mathematics in the Theory of Sets J. P. Mayberry Cambridge Cambridge University Press 2000 xx + 424 Hardback US$80.00 By J. P. Mayberry. Cambridge University Press. Cambridge. Pp. xx + 424. Hardback:US$80.00,

Stewart Shapiro. Introduction—intensional mathematics and constructive mathematics. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, vol. 113, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 1–10. - Stewart Shapiro. Epistemic and intuitionistic arithmetic. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 11–46. - John Myhill. Intensional set

Today the phrase “foundations of mathematics” has become synonymous with “set theory and mathematical logic.” The most important thing to understand about Kronecker’s views on the foundations of mathematics is that set theory and mathematical logic had no part in them.

I have argued elsewhere that second order logic provides a foundation for mathematics much in the same way as set theory does, despite the fact that the former is second order and the latter first order, but second order logic is marred by reliance on ad hoc {\em large domain assumptions}. In this paper I argue that sort logic, a powerful extension of second order logic, provides a foundation for mathematics without any ad hoc large domain assumptions. The large domain assumptions are replaced by ZFC-like axioms. Despite this resemblance to set theory sort logic retains the structuralist approach to mathematics characteristic of second order logic. As a model-theoretic logic sort logic is the strongest logic. In fact, every model class definable in set theory is the class of models of a sentence of sort logic. Because of its strength sort logic can be used to formulate particularly strong reflection principles in set theory.

J. P. Mayberry. The foundations of mathematics in the theory of sets. Encyclopedia of mathematics and its applications, vol. 82. Cambridge University Press, Cambridge 2000, New York 2001, etc., xx + 424 pp. - Volume 8 Issue 3

We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each other. However, our conclusion is that it is very difficult to see any real difference between the two. We analyze a phenomenonwe call internal categoricity which extends the familiar categoricity results of second order logic to Henkin models and show that set theory enjoys the same kind of internal categoricity. Thus the existence of non-standard models, which is usually taken as a property of first order set theory, and categoricity, which is usually taken as a property of second order axiomatizations, can coherently coexist when put into their proper context. We also take a fresh look at complete second order axiomatizations and give a hierarchy result for second order characterizable structures. Finally we consider the problem of existence in mathematics from both points of view and find that second order logic depends on what we call large domain assumptions, which come quite close to the meaning of the axioms of set theory.

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.

Kronecker's Views on the Foundations of Mathematics

Andrzej Mostowski. Recent results in set theory. Problems in the philosophy of mathematics, Proceedings of the International Colloquium in the Philosophy of Science, London, 1965, volume 1, edited by Imre Lakatos, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1967, pp. 82–96. - G. Kreisel, A. Robinson, L. Kalmár, and A. Mostowski. Discussion. Problems in the philosophy of mathematics, Proceedings of the International Colloquium in the Philosophy of Science, London, 1965, volume 1, edited by Imre Lakatos, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1967, pp. 97–108. - Andrzej Mostowski. O niektórych nowych wynikach meta-matematycznych dotyczących teorii mnogości (On some new metamathematical results concerning set theory). Polish with Russian and English summaries. Studia logica, vol. 20 (1967), pp. 99–116. - Volume 37 Issue 4

This article concentrates on exploring the relevance of the postmodernist concept of the event to mathematical philosophy and the foundations of mathematics. In both the scientific and philosophical study of nature, and particularly event ontology, we find that space and dynamism are fundamental. However, whether based on set theory or category theory, modern mathematics faces conceptual and philosophical difficulties when the temporal is intentionally invoked as a key aspect of that intrinsic dynamism so characteristic of mathematical being, physical becoming, process, and thought. We present a multidisciplinary investigation targeting a diverse audience including mathematicians, scientists, and philosophers who are interested in exploring alternative modes of doing mathematics or using mathematics to approach nature. Our aim is to understand both the formal character and the philosophy of time as realized through a radical mode of thinking that goes beyond the spatial in mathematics. In particular, we suggest the need to transcend the purely geometrical view altogether in future foundational research in both mathematics and mathematical philosophy. We reexamine these issues at a fundamental and comprehensive level, where a detailed exposition and critique of both modern set theories and theories of space is outlined, with emphasis on how the philosophy of Idealism has been permeating much of old and new mathematics. Furthermore, toward the end of the article, we explore some possible constructive directions in mathematical ontology by providing new proposals on how to develop a fragment of mathematics for the description of dynamic events.

This book offers a foundation for mathematics grounded in a collection of axioms for logical possibility in a first-order language. The offered foundation is argued to have various epistemological benefits, in particular as regards our justification for believing certain axioms of set theory, and more generally as regards the ‘access’ problem for mathematical objects. The foundation for mathematics is provided in stages. First, Berry argues for a ‘potentialist’ interpretation of standard ZF set theory in terms of some axioms for logical possibility that are argued to ‘seem clearly true’. Then, she argues for a Carnap-inspired account of the semantics of ordinary mathematical language in support of the idea that all mathematical discourse as usually pursued can be interpreted faithfully in terms of the foundation provided by her potentialist set theory. The end result, it is argued, is something approximating a logicist foundation for mathematics: standard mathematical truths are interpreted in terms of a pure logic of logical possibility.

Kenneth Kunen. Set theory. An introduction to independence proofs. Studies in logic and the foundations of mathematics, vol. 102. North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1980, xvi + 313 pp. - Volume 51 Issue 2