Axiomatizations of arithmetic and the first-order/second-order divide

I'm currently running a series of posts at M-Phi with sections of a paper I'm working on, 'Axiomatizations of arithmetic and the first-order/second-order divide', which may be of interest to at least some of the NewAPPS readership. It focuses on the idea that, when it comes to axiomatizing arithmetic, descriptive power and deductive power cannot be combined: axiomatizations that are categorical (using a highly expressive logical language, typically second-order logic) will typically be intractable, whereas axiomatizations with deductively better-behaved underlying logics (typically, first-order logic) will not be categorical -- i.e. will be true of models other than the intended model of the series of the natural numbers. Based on a distinction proposed by Hintikka between the descriptive use and the deductive use of logic in the foundations of mathematics, I discuss what the impossibility of having our arithmetical cake and eating it (i.e. of combining deductive power with expressive power to characterize arithmetic with logical tools) means for the first-order logic vs. second-order logic debate.

Part I is here, Part II here, and Part III here. I still hope to post Part IV tomorrow, and then the final Part V will have to wait for a while.