Nuno Maia

According to an extreme form of arithmetical pluralism, every consistent first-order extension of arithmetic has an intended model. Clarke-Doane objects that the view is committed to there being no objective fact as to whether arithmetic is consistent. He contends that the objection is avoided by retreating to a more restrictive form of arithmetical pluralism, according to which all and only Σ1-sound extensions of arithmetic have intended models. In this note, I argue that Clarke-Doane's solution is not successful since his original objection can be recreated with omega-consistent extensions of arithmetic. This result also shows that the objection is not avoided by appealing to Σn-consistency, against a recent suggestion to the contrary.

Theistic modal realism argues for an extension of Lewis's modal realism capable of accommodating a theistic God. By affording elegant solutions to many atheistic challenges, the view is of great theoretical utility for the theist. However, it has been objected that within a Lewisian framework God cannot be causally efficacious on pain of collapsing intuitively distinct modal notions. In this article I explain why these worries are ill-founded and show how God's existence and causal power over the pluriverse can be consistently understood. If successful, the proposal offers a congenial theistic way to adopt modal realism and address the atheological problems.

Boffa non-well-founded set theory allows for several distinct sets equal to their respective singletons, the so-called ‘Quine atoms’. Rieger contends that this theory cannot be a faithful description of set-theoretic reality. He argues that, even after granting that there are non-well-founded sets, ‘the extensional nature of sets’ precludes numerically distinct Quine atoms. In this paper we uncover important similarities between Rieger’s argument and how non-rigid structures are conceived within mathematical structuralism. This opens the way for an objection against Rieger, whilst affording the theoretical resources for a defence of Boffa set theory as a faithful description of set-theoretic reality.