Against the iterative conception of set

According to the iterative conception of set, each set is a collection of sets formed prior to it. The notion of priority here plays an essential role in explanations of why contradiction-inducing sets, such as the Russell set, do not exist. Consequently, these explanations are successful only to the extent that a satisfactory priority relation is made out. I argue that attempts to do this have fallen short: understanding priority in a straightforwardly constructivist sense threatens the coherence of the empty set and raises serious epistemological concerns; but the leading realist interpretations---ontological and modal interpretations of priority---are deeply problematic as well. I conclude that the purported explanatory virtues of the iterative conception are, at present, unfounded

The iterative conception of function and the iterative conception of set

Hilary Putnam once suggested that "the actual existence of sets as 'intangible objects' suffers... from a generalization of a problem first pointed out by Paul Benacerraf... are sets a kind of function or are functions a sort of set?" Sadly, he did not elaborate; my aim, here, is to do so on his behalf. There are well-known methods for treating sets as functions and functions as sets. But these do not raise any obvious philosophical or foundational puzzles. For that, we first need to provide a full-fledged function theory. I supply such a theory: it axiomatizes the iterative notion of function in exactly the same sense that ZF axiomatizes the iterative notion of set. Indeed, this function theory is synonymous with ZF. It might seem that set theory and function theory present us with rival foundations for mathematics, since they postulate different ontologies. But appearances are deceptive. Set theory and function theory provide the very same judicial foundation for mathematics. They do not supply rival metaphysical foundations; indeed, if they supply metaphysical foundations at all, then they supply the very same metaphysical foundations

Large Cardinals and the Iterative Conception of Set

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. One idea sometimes alluded to is that maximality considerations speak in favour of large cardinal axioms consistent with ZFC, since it appears to be `possible' (in some sense) to continue the hierarchy far enough to generate the relevant transfinite number. In this paper, we argue against this idea based on a priority of subset formation under the iterative conception. In particular, we argue that there are several conceptions of maximality that justify the consistency but falsity of large cardinal axioms. We argue that the arguments we provide are illuminating for the debate concerning the justification of new axioms in iteratively-founded set theory

Large Cardinals and the Iterative Conception of Set

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. One idea sometimes alluded to is that maximality considerations speak in favour of large cardinal axioms consistent with ZFC, since it appears to be `possible' (in some sense) to continue the hierarchy far enough to generate the relevant transfinite number. In this paper, we argue against this idea based on a priority of subset formation under the iterative conception. In particular, we argue that there are several conceptions of maximality that justify the consistency but falsity of large cardinal axioms. We argue that the arguments we provide are illuminating for the debate concerning the justification of new axioms in iteratively-founded set theory

Large Cardinals and the Iterative Conception of Set

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles for the iterative conception, and assert that the length of the iterative stages is as long as possible. In this paper, we argue that whether or not large cardinal principles count as maximality principles depends on prior commitments concerning the richness of the subset forming operation. In particular we argue that there is a conception of maximality through absoluteness, that when given certain technical formulations, supports the idea that large cardinals are consistent, but false. On this picture, large cardinals are instead true in inner models and serve to restrict the subsets formed at successor stages

Quantification and Paradox

I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantification, but on the logical conception of set which motivates naive set theory. The accepted solution is to replace this with the iterative conception of set. I show that this picture is doubly mistaken. After a close examination of the paradoxes in chapters 2--3, I argue in chapters 4 and 5 that it is possible to rescue naive set theory by restricting quantification over sets and that the resulting restrictivist set theory is expressible. In chapters 6 and 7, I argue that it is the iterative conception of set and the thesis of absolutism that should be rejected

Large Cardinals and the Iterative Conception of Set

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles for the iterative conception, and assert that the length of the iterative stages is as long as possible. In this paper, we argue that whether or not large cardinal principles count as maximality principles depends on prior commitments concerning the richness of the subset forming operation. In particular we argue that there is a conception of maximality through absoluteness, that when given certain technical formulations, supports the idea that large cardinals are consistent, but false. On this picture, large cardinals are instead true in inner models and serve to restrict the subsets formed at successor stages

Why Is the Universe of Sets Not a Set?

According to the iterative conception of sets, standardly formalized by ZFC, there is no set of all sets. But why is there no set of all sets? A simple-minded, though unpopular, "minimal" explanation for why there is no set of all sets is that the supposition that there is contradicts some axioms of ZFC. In this paper, I first explain the core complaint against the minimal explanation, and then argue against the two main alternative answers to the guiding question. I conclude the paper by outlining a close alternative to the minimal explanation, the conception-based explanation, that avoids the core complaint against the minimal explanation

Why Is the Universe of Sets Not a Set?

According to the iterative conception of sets, standardly formalized by ZFC, there is no set of all sets. But why is there no set of all sets? A simple-minded, though unpopular, "minimal" explanation for why there is no set of all sets is that the supposition that there is contradicts some axioms of ZFC. In this paper, I first explain the core complaint against the minimal explanation, and then argue against the two main alternative answers to the guiding question. I conclude the paper by outlining a close alternative to the minimal explanation, the conception-based explanation, that avoids the core complaint against the minimal explanation