The shortest definition of a number by a first order formula with one free variable, where the notion of a formula defining a number extends a notion used by Boolos in a proof of the Incompleteness Theorem, is shown to be non computable. This is followed by an examination of the complexity of sets associated with this function.

This paper intervenes in an argument over the number of thoughts that could be thought. The argument has important implications for supervenience physicalism, the thesis that all is physical or supervenient on the physical. If, per quantum mechanics, the number of possible physical states is finite while the number of possible thoughts is infinite, then the latter exceeds the former in number, and supervenience phyicalsim fails. Abelson first argued that possible thoughts are infinite as we can think of any of (...) the infinite natural numbers. Subsequently, physicist Max Tegmark argued that we cannot think of all the numbers there are, that some are simultaneously too large and too nondescript to reference. Porpora offered a brief proof countering Tegmark. Curtis then countered Porpora, arguing that Porpora’s argument runs afoul of the Berry paradox. This paper shows that while Curtis does offer an analogous proof that does fall prey to the Berry paradox, Porpora’s does not. The result reinstates Porpora’s argument with all its implications for supervenience physicalism and offers a clearer lesson from the Berry Paradox. (shrink)

This paper is a response to Haze’s brief argument for the falsity of the theory that instantial terms refer arbitrarily, proposed by Breckenridge and Magidor in 2012. In this paper, I characterise instantial terms and outline the theory of arbitrary reference; then I reconstruct Haze’s argument and contend that it fails in its purpose. Haze’s argument is supposed to be a _reductio ad absurdum:_ according to Haze, it proves that a contradiction follows from the most basic tenets of the theory (...) of arbitrary reference. I will argue, however, that the contradiction in question follows not from these tenets, but from the surreptitious use that Haze makes of a self-referential expression. I conclude, consequently, that Haze’s argument is nothing more than an illustration of the well-known fact that self-referential expressions produce paradoxical results. (shrink)