An engaging account of the origins of the modern mathematical theory of the infinite, his book is also a spirited defense against the attacks and misconceptions ...

The view that plural reference is reference to a set is examined in light of George Boolos's treatment of second-order quantification as plural quantification in English. I argue that monadic second-order logic does not, in Boolos's treatment, reflect the behavior of plural quantifiers under negation and claim that any sentence that properly translates a second-order formula, in accordance with his treatment, has a first-order formulation. Support for this turns on the use of certain partitive constructions to assign values to variables (...) in a way that makes Boolos's reading of second-order variables available for a first-order language and, with it, the possibility of interpreting quantification in an unrestricted domain.A first-order theory, T(D), is developed on the basis of Boolos's treatment of simple plural definite descriptions extended to Richard Sharvy's general theory of definite plural and mass descriptions. I introduce a primitive predicate, o, for the relation of the referent of a singular description to that of its plural. If o is simply added to T(D), is definable in T(D), and the result is inconsistent. If o is added to a theory with axioms for the fragment of T(D) I call D-mereology, the result is a natural basis for the development of a pluralized Zermelo set theory. This theory, however, is inconsistent in an unrestricted domain, unless it is recast as a second-order theory of sets interpreted in Boolos's way. (shrink)

A new axiomatization of set theory, to be called Bernays-Boolos set theory, is introduced. Its background logic is the plural logic of Boolos, and its only positive set-theoretic existence axiom is a reflection principle of Bernays. It is a very simple system of axioms sufficient to obtain the usual axioms of ZFC, plus some large cardinals, and to reduce every question of plural logic to a question of set theory.

According to structuralism in philosophy of mathematics, arithmetic is about a single structure. First-order theories are satisfied by models that do not instantiate this structure. Proponents of structuralism have put forward various accounts of how we succeed in fixing one single structure as the intended interpretation of our arithmetical language. We shall look at a proposal that involves Tennenbaum's theorem, which says that any model with addition and multiplication as recursive operations is isomorphic to the standard model of arithmetic. On (...) this account, the intended models of arithmetic are the notation systems with recursive operations on them satisfying the Peano axioms. [A]m Anfang […] ist das Zeichen. (shrink)

The major point of contention among the philosophers and mathematicians who have written about the independence results for the continuum hypothesis (CH) and related questions in set theory has been the question of whether these results give reason to doubt that the independent statements have definite truth values. This paper concerns the views of G. Kreisel, who gives arguments based on second order logic that the CH does have a truth value. The view defended here is that although Kreisel's conclusion (...) is correct, his arguments are unsatisfactory. Later sections of the paper advance a different argument that the independence results do not show lack of truth values. (shrink)

When viewed as the most comprehensive theory of collections, set theory leaves no room for classes. But the vocabulary of classes, it is argued, provides us with compact and, sometimes, irreplaceable formulations of largecardinal hypotheses that are prominent in much very important and very interesting work in set theory. Fortunately, George Boolos has persuasively argued that plural quantification over the universe of all sets need not commit us to classes. This paper suggests that we retain the vocabulary of classes, but (...) explain that what appears to be singular reference to classes is, in fact, covert plural reference to sets. (shrink)

The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify (...) many concepts in contemporary mathematics, and thus that both first- and higher-order logics are needed to fully reflect current work. Throughout, the emphasis is on discussing the associated philosophical and historical issues and the implications they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic comparable to that provided in a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in the field today. (shrink)

This note examines the mereological component of Geoffrey Hellman's most recent version of modal structuralism. There are plausible forms of agnosticism that benefit only a little from Hellman's mereological turn.

This essay defends the following two claims: (1) liraitation-of-size reasoning yields enough sets to meet the needs of most mathematicians; (2) set formation and mereological fusion share enough logical features to justify placing both in the genus composition (even when the components of a set are taken to be its members rather than its subsets).

Arecent paper by George Boolos suggests that it is philosophically respectable to use monadic second order logic in one’s explication of the iterative concept of set. I shall here give a partial indication of the new range of theories of the iterative hierarchy which are thus madeavailable to philosophers of set theory.

There is no preferred reduction of number theory to set theory. Nonetheless, we confidently accept axioms obtained by substituting formulas from the language of set theory into the induction axiom schema. This is only possible, it is argued, because our acceptance of the induction axioms depends solely on the meanings of aritlunetical and logical terms, which is only possible if our 'intended models' of number theory are standard. Similarly, our acceptance of the second-order natural deduction rules depends solely on the (...) meanings of the logical terms, which implies, it is argued, that our second-order quantifiers have to be standard. (shrink)

Mathematical realism is the doctrine that mathematical objects really exist, that mathematical statements are either determinately true or determinately false, and that the accepted mathematical axioms are predominantly true. A realist understanding of set theory has it that when the sentences of the language of set theory are understood in their standard meaning, each sentence has a determinate truth value, so that there is a fact of the matter whether the cardinality of the continuum is א2 or whether there are (...) measurable cardinals, whether or not those facts are knowable by us. (shrink)

This paper criticizes George Boolos's famous use of plural quantification to argue that monadic second-order logic is pure logic. I deny that plural quantification qualifies as pure logic and express serious misgivings about its alleged ontological innocence. My argument is based on an examination of what is involved in our understanding of the impredicative plural comprehension schema.

is the second-order theory of the part-whole relation. It can express such hypotheses about the size of Reality as that there are inaccessibly many atoms. Take a non-empty class to have exactly its non-empty subclasses as parts; hence, its singleton subclasses as atomic parts. Then standard set theory becomes the theory of the member-singleton function—better, the theory of all singleton functions—within the framework of megethology. Given inaccessibly many atoms and a specification of which atoms are urelements, a singleton function exists, (...) unique up to isomorphism. (This article is partly abridged from my Parts of Classes, partly a sequel.). (shrink)

This paper explores strengths and limitations of both predicativism and nominalism, especially in connection with the problem of characterizing the continuum. Although the natural number structure can be recovered predicatively (despite appearances), no predicative system can characterize even the full predicative continuum which the classicist can recognize. It is shown, however, that the classical second-order theory of continua (third-order number theory) can be recovered nominalistically, by synthesizing mereology, plural quantification, and a modal-structured approach with essentially just the assumption that an (...) -sequence of concrete atoms be possible. Predicative flexible type theory may then be used to carry out virtually all of scientifically applicable mathematics in a natural way, still without ultimate need of the platonist ontology of classes and relations. (shrink)

Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis-a-vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell's many-topoi view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out (...) relative to such domains; puzzles about ‘large categories’ and ‘proper classes’ are handled in a uniform way, by relativization, sustaining insights of Zermelo. (shrink)

I examine various claims to the effect that Cantor's Continuum Hypothesis and other problems of higher set theory are ill-posed questions. The analysis takes into account the viability of the underlying philosophical views and recent mathematical developments.

Presenting a selection of thirteen essays on various topics at the foundations of philosophy--one previously unpublished and eight accompanied by substantial new postscripts--this book offers outstanding insight on truth, meaning, and propositional attitudes; semantic indeterminacy and other kinds of "factual defectiveness;" and issues concerning objectivity, especially in mathematics and in epistemology. It will reward the attention of any philosopher interested in language, epistemology, or mathematics.

Part of the ambiguity lies in the various points of view from which this question might be considered. The crudest di erence lies between the point of view of the working mathematician and that of the logician concerned with the foundations of mathematics. Now some of my fellow mathematical logicians might protest this distinction, since they consider themselves to be just more of those \working mathematicians". Certainly, modern logic has established itself as a very respectable branch of mathematics, and there (...) are quite a few highly technical journals in logic, such as The Journal of Sym-. (shrink)

Mathematical realism is the doctrine that mathematical objects really exist, that mathematical statements are either determinately true or determinately false, and that the accepted mathematical axioms are predominantly true. A realist understanding of set theory has it that when the sentences of the language of set theory are understood in their standard meaning, each sentence has a determinate truth value, so that there is a fact of the matter whether the cardinality of the continuum is א2 or whether there are (...) measurable cardinals, whether or not those facts are knowable by us. (shrink)

NOWADAYS, even schoolchildren babble about "null sets" and "singletons" and "one-one correspondences," as if they knew what they were talking about. But if they understand even less than their teachers, which seems likely, they must be using the technical jargon with only an illusion of understanding. Beginners are taught that a set having three members is a single thing, wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as "three in one" should (...) be child's play. (shrink)