A Note On Clarke-Doane’s Soundness Requirement

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 ∑1-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 ∑n-consistency, against a recent suggestion to the contrary.