Bi-Modal Naive Set Theory

This paper describes a modal conception of sets, according to which sets are 'potential' with respect to their members. A modal theory is developed, which invokes a naive comprehension axiom schema, modified by adding 'forward looking' and 'backward looking' modal operators. We show that this 'bi-modal' naive set theory can prove modalized interpretations of several ZFC axioms, including the axiom of infinity. We also show that the theory is consistent by providing an S5 Kripke model. The paper concludes with some discussion of the nature of the modalities involved, drawing comparisons with noneism, the view that there are some non-existent objects

Metaphysical Dependence and Set Theory

In this dissertation, I articulate and defend a counterfactual analysis of metaphysical dependence. It is natural to think that one thing x depends on another thing y if had y not existed, then x wouldn\u27t have existed either. But counterfactual analyses of metaphysical dependence are often rejected in the current literature. They are rejected because straightforward counterfactual analyses fail to accurately capture dependence relations between objects that exist necessarily, like mathematical objects. For example, it is taken as given that sets metaphysically depend on their members, while members do not metaphysically depend on the sets they belong to. The set {0} metaphysically depends on 0, while 0 does not metaphysically depend on {0}. The dependence is asymmetric. But if counterfactuals are given a possible worlds analysis, as is standard, then the counterfactual approach to dependence will yield a symmetric dependence relation between these two sets. Because the counterfactual analysis fails to accurately capture dependence relations between sets and their members, most reject this approach to metaphysical dependence. To generate the desired asymmetry, I argue that we should introduce impossible worlds into the framework for evaluating counterfactuals. I review independent reasons for admitting impossible worlds alongside possible worlds. Once we have impossible worlds at our disposal, we can consider worlds where, e.g., the empty set does not exist. I argue that in the worlds that are ceteris paribus like the actual world, where 0 does not exist, {0} does not exist either. And so, according to the counterfactual analysis of dependence, {0} metaphysically depends on 0, as desired. Conversely, however, there is no reason to think that every world that is ceteris paribus like the actual world, where {0} does not exist, is such that 0 does not exist either. And so 0 does not metaphysically depend on {0}. After applying this extended counterfactual analysis to several set-theoretic cases, I show that it can be applied to account for dependence relations between other mathematical objects as well. I conclude by defending the counterfactual analysis, extended with impossible worlds, against several objections

The Structuralist Thesis Reconsidered

Øystein Linnebo and Richard Pettigrew ([2014]) have recently developed a version of non-eliminative mathematical structuralism based on Fregean abstraction principles. They argue that their theory of abstract structures proves a consistent version of the structuralist thesis that positions in abstract structures only have structural properties. They do this by defining a subset of the properties of positions in structures, so-called fundamental properties, and argue that all fundamental properties of positions are structural. In this paper, we argue that the structuralist thesis, even when restricted to fundamental properties, does not follow from the theory of structures that Linnebo and Pettigrew have developed. To make their account work, we propose a formal framework in terms of Kripke models that makes structural abstraction precise. The formal framework allows us to articulate a revised definition of fundamental properties, understood as intensional properties. Based on this revised definition, we show that the restricted version of the structuralist thesis holds

The Structuralist Thesis Reconsidered

Øystein Linnebo and Richard Pettigrew ([2014]) have recently developed a version of non-eliminative mathematical structuralism based on Fregean abstraction principles. They argue that their theory of abstract structures proves a consistent version of the structuralist thesis that positions in abstract structures only have structural properties. They do this by defining a subset of the properties of positions in structures, so-called fundamental properties, and argue that all fundamental properties of positions are structural. In this paper, we argue that the structuralist thesis, even when restricted to fundamental properties, does not follow from the theory of structures that Linnebo and Pettigrew have developed. To make their account work, we propose a formal framework in terms of Kripke models that makes structural abstraction precise. The formal framework allows us to articulate a revised definition of fundamental properties, understood as intensional properties. Based on this revised definition, we show that the restricted version of the structuralist thesis holds

Set-Theoretic Dependence

In this paper, we explore the idea that sets depend on, or are grounded in, their members. It is said that a set depends on each of its members, and not vice versa. Members do not depend on the sets that they belong to. We show that the intuitive modal truth conditions for dependence, given in terms of possible worlds, do not accurately capture asymmetric dependence relations between sets and their members. We extend the modal truth conditions to include impossible worlds and give a more satisfactory account of the dependence of a set on its members. Focusing on the case of singletons, we articulate a logical framework in which to evaluate set-theoretic dependence claims, using a normal first-order modal logic. We show that on this framework the dependence of a singleton on its single members follows from logic alone. However, the converse does not hold

Bi-Modal Naive Set Theory

This paper describes a modal conception of sets, according to which sets are 'potential' with respect to their members. A modal theory is developed, which invokes a naive comprehension axiom schema, modified by adding 'forward looking' and 'backward looking' modal operators. We show that this 'bi-modal' naive set theory can prove modalized interpretations of several ZFC axioms, including the axiom of infinity. We also show that the theory is consistent by providing an S5 Kripke model. The paper concludes with some discussion of the nature of the modalities involved, drawing comparisons with noneism, the view that there are some non-existent objects

Geology and geochemistry of the Almanor and Glidden barite deposits, northern California

Online access for this thesis was created in part with support from the Institute of Museum and Library Services (IMLS) administered by the Nevada State Library, Archives and Public Records through the Library Services and Technology Act (LSTA). To obtain a high quality image or document please contact the DeLaMare Library at https://unr.libanswers.com/ or call: 775-784-6945.The Almanor and Glidden bedded barite deposits are at the northern end of a 700 km long "California barite belt", and are associated with siliceous sediments and volcanic arc derived rocks of Devonian (?) and middle Devonian age respectively. Regional geologic relations allow correlation of these deposits with similar deposits of north central Nevada. Geochemical evidence suggests a hydrothermal source for the barium. Cyclic stratigraphic variations in the strontium content of barite from the Glidden deposit may be the result of cyclic temporal variations in the physico-chemical environment of mineralization