A number of formal constraints on acceptable abstraction principles have been proposed, including conservativeness and irenicity. Hume’s Principle, of course, satisfies these constraints. Here, variants of Hume’s Principle that allow us to count concepts instead of objects are examined. It is argued that, prima facie, these principles ought to be no more problematic than HP itself. But, as is shown here, these principles only enjoy the formal properties that have been suggested as indicative of acceptability if certain constraints on the (...) size of the continuum hold. As a result, whether or not these higher-order versions of Hume’s Principle are acceptable seems to be independent of standard (ZFC) set theory. This places the abstractionist in an uncomfortable dilemma: Either there is some inherent difference between counting objects and counting concepts, or new criteria for acceptability will need to be found. It is argued that neither horn looks promising. (shrink)

In [Boccuni 2010], a predicative fragment of Frege’s blv augmented with Boolos’ unrestricted plural quantification is shown to interpret pa2. The main disadvantage of that axiomatisation is that it does not recover Frege Arithmetic fa because of the restrictions imposed on the axioms. The aim of the present article is to show how [Boccuni 2010] can be consistently extended so as to interpret fa and consequently pa2 in a way that parallels Frege’s. In that way, the presented system will be (...) compared with the system pe in [Ferreira 2018] and some relevant differences between the two will be highlighted. (shrink)

In [Boccuni 2010], a predicative fragment of Frege’s blv augmented with Boolos’ unrestricted plural quantification is shown to interpret pa2. The main disadvantage of that axiomatisation is that it does not recover Frege Arithmetic fa because of the restrictions imposed on the axioms. The aim of the present article is to show how [Boccuni 2010] can be consistently extended so as to interpret fa and consequently pa2 in a way that parallels Frege’s. In that way, the presented system will be (...) compared with the system pe in [Ferreira 2018] and some relevant differences between the two will be highlighted. (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)

The paper is concerned with the bad company problem as an instance of a more general difficulty in the philosophy of mathematics. The paper focuses on the prospects of stability as a necessary condition on acceptability. However, the conclusion of the paper is largely negative. As a solution to the bad company problem, stability would undermine the prospects of a neo-Fregean foundation for set theory, and, as a solution to the more general difficulty, it would impose an unreasonable constraint on (...) mathematical practice. (shrink)

1. Philosophical background: iteration, ineffability, reflection. There are at least two heuristic motivations for the axioms of standard set theory, by which we mean, as usual, first-order Zermelo–Fraenkel set theory with the axiom of choice (ZFC): the iterative conception and limitation of size (see Boolos, 1989). Each strand provides a rather hospitable environment for the hypothesis that the set-theoretic universe is ineffable, which is our target in this paper, although the motivation is different in each case.

Objects appear to fall into different sorts, each with their own criteria for identity. This raises the question of whether sorts overlap. Abstractionists about numbers—those who think natural numbers are objects characterized by abstraction principles—face an acute version of this problem. Many abstraction principles appear to characterize the natural numbers. If each abstraction principle determines its own sort, then there is no single subject-matter of arithmetic—there are too many numbers. That is, unless objects can belong to more than one sort. (...) But if there are multi-sorted objects, there should be cross-sortal identity principles for identifying objects across sorts. The going cross-sortal identity principle, ECIA2 of Cook and Ebert (2005), solves the problem of too many numbers. But, I argue, it does so at a high cost. I therefore propose a novel cross-sortal identity principle, based on embeddings of the induced models of abstracts developed by Walsh (2012). The new criterion matches ECIA2’s success, but offers interestingly different answers to the more controversial identifications made by ECIA2. (shrink)