We define a class of functions, the descent recursive functions, relative to an arbitrary elementary recursive system of ordinal notations. By means of these functions, we provide a general technique for measuring the proof-theoretic strength of a variety of systems of first-order arithmetic. We characterize the provable well-orderings and provably recursive functions of these systems, and derive various conservation and equiconsistency results.

A subtheory of the theory of self-referential truth known as FS is shown to be weak as a theory of truth but equivalent to full FS in its proof-theoretic strength.

How do we weigh the claims of current and future people when current exercise of rights to property conflict with sustainability? Are property rights over theseresources more limited due to the claims of posterity? Lockean property rights allow no right to degrade resources when doing so threatens the basic needs offuture generations. A stewardship conception of property rights can be developed, providing a justification for sustainable management legislation even whensuch law conflicts with the rights an owner would have, were the (...) resource under more full-blown ownership. A protection indicator can be developed that is sensitive to a range of empirical factors such as scarcity, renewability, importance of the resource, and seriousness and reversibility of potential harm. The stewardship conception of rights over environmental resources can be applied in policy settings, for example, in decisions over emissions limits and land-use patterns. Such harnessing of Lockean intuitions to argue for environmental protection is in sharp contrast to Locke’s usual employment by those keen to show that such protection violates owners’ rights. (shrink)

In a modal system of arithmetic, a theory S has the modal disjunction property if whenever $S \vdash \square\varphi \vee \square\psi$ , either $S \vdash \square\varphi$ or $S \vdash \square\psi. S$ has the modal numerical existence property if whenever $S \vdash \exists x\square\varphi(x)$ , there is some natural number n such that $S \vdash \square\varphi(\mathbf{n})$ . Under certain broadly applicable assumptions, these two properties are equivalent.

Probably the two most famous examples of elementary embeddings between inner models of set theory are the embeddings of the universe into an inner model given by a measurable cardinal and the embeddings of the constructible universeLinto itself given by 0#. In both of these examples, the “target model” is a subclass of the “ground model”. It is not hard to find examples of embeddings in which the target model is not a subclass of the ground model: ifis a generic (...) ultrafilter arising from forcing with a precipitous ideal on a successor cardinalκ, then the ultraproduct of the ground model viacollapsesκ. Such considerations suggest a classification of how close the target model comes to “fitting inside” the ground model.Definition 1.1. LetMandNbe inner models of ZFC, and letj:M→Nbe an elementary embedding. Theco-critical pointofjis the least ordinalλ, if any exist, such that there isX⊆λ, X∈NbutX∉M. Such anXis called anew subsetofλ.It is easy to see that the co-critical point ofj:M→Nis a cardinal inN. (shrink)

Businesses interface with the natural world through rights to property. The shape of these rights and the responsibilities we assign to managers are important determinants of both patterns of resource use and pollutant levels. Consequently, conflicts have arisen between regulating bodies, indigenous groups, and corporations over the entitlements of businesses in the use of their property when that property is ecologically sensitive or significant. In this paper I develop an account of the ethical responsibilities of managers regarding their treatment of (...) the environment and their use of natural resources. This account is based on a philosophical examination of the nature of property rights. After a tour of traditional arguments employed to defend the institution of private property, I develop a new conception of property rights over objects that have a high natural resource component. I show that in today’s world, ethical concerns that motivated John Locke will yield property rights that are insufficiently strong to override a democratic say in overall use patterns and must be sensitive to factors such as need, scarcity, and the interests of stakeholders. I show a similar result for utilitarian arguments for property. On this foundation I develop a conception of property in key environmental resources that includes a stewardship element. To embody this, I suggest a set of ethical responsibilities for managers over certain natural resources that springs from it. I argue for obligations restricting harmful use patterns as a way of including the other goals (eg public participation, sustainability) that we care about, in a manner which clashes least with the liberty of owners and the efficiency of production. The outcome is a set of rights and responsibilities that take global environmental interests into account, but preserve the thrust of what is good about the institution of private property: that it rewards labour and facilitates a stable and efficient free exchange system. As a version of private property, it retains many of the advantages of global capitalism while attaching responsibilities to business decision makers. (shrink)

In a language for arithmetic with a predicate T, intended to mean “ x is the Gödel number of a true sentence”, a set S of axioms and rules of inference has the truth disjunction property if whenever S ⊢ T ∨ T, either S ⊢ T or S ⊢ T. Similarly, S has the truth existence property if whenever S ⊢ ∃χ T ), there is some n such that S ⊢ T ). Continuing previous work, we establish whether (...) these properties hold or fail for a large collection of possible axiomatic systems. (shrink)