This paper is a reply to George Boolos's three papers (Boolos (1987a, 1987b, 1990a)) concerned with the status of Hume's Principle. Five independent worries of Boolos concerning the status of Hume's Principle as an analytic truth are identified and discussed. Firstly, the ontogical concern about the commitments of Hume's Principle. Secondly, whether Hume's Principle is in fact consistent and whether the commitment to the universal number by adopting Hume's Principle might be problematic. Also the so-called `surplus content' worry is discussed, (...) which points out that the conceptual resources to grasp Hume's Principle vastly outstrip the conceptual resources employed in arithmetical reasoning. And lastly whether Hume's Principle is in bad company with other unsuccessful implicit definitions. In the last section, an account towards our entitlement to Hume&'s Principle is sketched. (shrink)

One recent `neologicist' claim is that what has come to be known as "Frege's Theorem"–the result that Hume's Principle, plus second-order logic, suffices for a proof of the Dedekind-Peano postulate–reinstates Frege's contention that arithmetic is analytic. This claim naturally depends upon the analyticity of Hume's Principle itself. The present paper reviews five misgivings that developed in various of George Boolos's writings. It observes that each of them really concerns not `analyticity' but either the truth of Hume's Principle or our entitlement (...) to accept it and reviews possible neologicist replies. A two-part Appendix explores recent developments of the fifth of Boolos's objections–the problem of Bad Company–and outlines a proof of the principle $N^q$, an important part of the defense of the claim that what follows from Hume's Principle is not merely a theory which allows of interpretation as arithmetic but arithmetic itself. (shrink)

Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, we show (...) that most other abstraction principles are not naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to stability-like acceptability criteria on abstraction principles, the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli and Fine, and supervaluational ideas coming out of Hodes' work. (shrink)

Frege's Grundgesetze was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem shows that there is a model of a fragment of the Grundgesetze which defines a (...) model of all the axioms of Zermelo-Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to G\"odel's constructible universe of sets, which G\"odel famously used to show the relative consistency of the continuum hypothesis. More specifically, our proofs appeal to Kripke and Platek's idea of the projectum within the constructible universe as well as to a weak version of uniformization (which does not involve knowledge of Jensen's fine structure theory). The axioms of the Grundgesetze are examples of abstraction principles, and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension. As an application, we resolve an analogue of the joint consistency problem in the predicative setting. (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)

Neologicists have sought to ground mathematical knowledge in abstraction. One especially obstinate problem for this account is the bad company problem. The leading neologicist strategy for resolving this problem is to attempt to sift the good abstraction principles from the bad. This response faces a dilemma: the system of ‘good’ abstraction principles either falls foul of the Scylla of inconsistency or the Charybdis of being unable to recover a modest portion of Zermelo–Fraenkel set theory with its intended generality. This article (...) argues that the bad company problem is due to the ‘static’ character of abstraction on the neologicist's account and develops a ‘dynamic’ account of abstraction that avoids both horns of the dilemma. 1 Introduction2ion Reconstructed3 From Bad to Worse4 Abstraction Reconceived5 Bad Company Made GoodA AppendixA.1SemanticsA.2LogicA.3ConsistencyA.4Strength. (shrink)

A number of authors have recently weighed in on the issue of whether it is coherent to have bound variables that range over absolutely everything. Prima facie, it is difficult, and perhaps impossible, to coherently state the “relativist” position without violating it. For example, the relativist might say, or try to say, that for any quantifier used in a proposition of English, there is something outside of its range. What is the range of this quantifier? Or suppose we ask the (...) relativist if there are some things that cannot appear in the range of any bound variable. The likely response would be along these lines: “No. For each object o, it possible to include o in the range of quantifiers, but one cannot quantify over everything at once.” This sentence contains unrestricted quantifiers, or so it seems, pending some clever move from a relativist. On the other hand, in the context of set theory, the reasoning behind the Burali-Forti paradox strongly suggests that there are well-orderings strictly longer than the collection of all ordinals. And set theorists regularly do transfinite recursions and transfinite reductions along such well-orderings. The relativist simply points out that one can always define new ordinals, and thus expand the range of one’s bound variables. The purpose of this paper is to explore the iterative framework, proposed in Zermelo’s 1930 paper, “Über Grenzzahlen und Mengenbereiche” (“On boundary numbers and domains of sets”), in order to shed light on these issues, and see what is involved in resolving them. (shrink)

We examine George Boolos's proposed abstraction principle for extensions based on the limitation-of-size conception, New V, from several perspectives. Crispin Wright once suggested that New V could serve as part of a neo-logicist development of real analysis. We show that it fails both of the conservativeness criteria for abstraction principles that Wright proposes. Thus, we support Boolos against Wright. We also show that, when combined with the axioms for Boolos's iterative notion of set, New V yields a system equivalent to (...) full Zermelo-Fraenkel set theory with a principle of global choice. This advances Boolos's longstanding interest in the foundations of set theory. (shrink)

I offer in this paper a contextual analysis of Frege's Grundlagen, section 64. It is surprising that with so much ink spilled on that section, the sources of Frege's discussion of definitions by abstraction have remained elusive. I hope to have filled this gap by providing textual evidence coming from, among other sources, Grassmann, Schlömilch, and the tradition of textbooks in geometry for secondary schools . In addition, I put Frege's considerations in the context of a widespread debate in Germany (...) on ‘directions’ as a central notion in the theory of parallels. (shrink)

If numbers were identified with any of their standard set-theoretic realizations, then they would have various non-arithmetical properties that mathematicians are reluctant to ascribe to them. Dedekind and later structuralists conclude that we should refrain from ascribing to numbers such ‘foreign’ properties. We first rehearse why it is hard to provide an acceptable formulation of this conclusion. Then we investigate some forms of abstraction meant to purge mathematical objects of all ‘foreign’ properties. One form is inspired by Frege; the other (...) by Dedekind. We argue that both face problems. (shrink)

Frege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be one-to-one correlated. According to Frege’s Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying second-order logic—and (...) investigates how much of Frege’s Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension. (shrink)

Say that some things form a set just in case there is a set whose members are precisely the things in question. For instance, all the inhabitants of New York form a set. So do all the stars in the universe. And so do all the natural numbers. Under what conditions do some things form a set?

In this paper I explore a position on which it is possible to eliminate the need for postulating abstract objects through abstraction principles by treating terms for abstracta as ‘incomplete symbols’, using Russell's no-classes theory as a template from which to generalize. I defend views of this stripe against objections, most notably Richard Heck's charge that syntactic forms of nominalism cannot correctly deal with non-first-orderizable quantifcation over apparent abstracta. I further discuss how number theory may be developed in a system (...) treating apparent terms for numbers using these definitions. (shrink)

After briefly reviewing the standard set-theoretic resolutions of the Burali-Forti paradox, we examine how the paradox arises in set theory formalized with plural quantifiers. A significant choice emerges between the desirable unrestricted availability of ordinals to represent well-orderings and the sensibility of attempting to refer to ‘absolutely all ordinals’ or ‘absolutely all well-orderings’. This choice is obscured by standard set theories, which rely on type distinctions which are obliterated in the setting with plurals. Zermelo's attempt ( 1930 ) to secure (...) ordinal representability of arbitrary well-orderings through relativization of quantification to set-theoretic models is reviewed and found wanting. The natural modal-structural recasting provides, it is claimed, a good repair. (shrink)

As is well-known, the formal system in which Frege works in his Grundgesetze der Arithmetik is formally inconsistent, Russell’s Paradox being derivable in it.This system is, except for minor differ...

This paper argues for the thesis that, roughly put, it is impossible to talk about absolutely everything. To put the thesis more precisely, there is a particular sense in which, as a matter of semantics, quantifiers always range over domains that are in principle extensible, and so cannot count as really being ‘absolutely everything’. The paper presents an argument for this thesis, and considers some important objections to the argument and to the formulation of the thesis. The paper also offers (...) an assessment of just how implausible the thesis really is. It argues that the intuitions against the thesis come down to a few special cases, which can be given special treatment. Finally, the paper considers some metaphysical ideas that might surround the thesis. Particularly, it might be maintained that an important variety of realism is incompatible with the thesis. The paper argues that this is not the case. (shrink)

It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full (i.e., with unrestricted comprehension) second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae. He further conjectured that the more encompassing Δ₁¹-comprehension (...) schema would already be inconsistent. In the present paper, we show that this is not the case. (shrink)

It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full (i.e., with unrestricted comprehension) second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae. He further conjectured that the more encompassing Δ11-comprehension (...) schema would already be inconsistent. In the present paper, we show that this is not the case. (shrink)

Neo-Fregeans have been troubled by the Nuisance Principle, an abstraction principle that is consistent but not jointly satisfiable with the favored abstraction principle HP. We show that logically this situation persists if one looks at joint consistency rather than satisfiability: under a modest assumption about infinite concepts, NP is also inconsistent with HP.

A neologicist set theory based on an abstraction principle (NewerV) codifying the iterative conception of set is investigated, and its strength is compared to Boolos's NewV. The new principle, unlike NewV, fails to imply the axiom of replacement, but does secure powerset. Like NewV, however, it also fails to entail the axiom of infinity. A set theory based on the conjunction of these two principles is then examined. It turns out that this set theory, supplemented by a principle stating that (...) there are infinitely many nonsets, captures all (or enough) of standard second-order ZFC. Issues pertaining to the axiom of foundation are also investigated, and I conclude by arguing that this treatment provides the neologicist with the most viable reconstruction of set theory he is likely to obtain. (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.