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    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.

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    2 responses to “Axiomatizations of arithmetic and the first-order/second-order divide”

    1. Jon Cogburn Avatar
      Jon Cogburn

      Thanks so much for this! I’ve just started reading, but I have a probably too simple question early on. Consider:

      It is often argued (as discussed in (Rossberg 2004)) that the problematic status of the second-order consequence relation is sufficient to exclude second-order logic from the realm of what counts as ‘logic’.

      And, if I understand right, your response is that this involves privileging deduction too much.
      Here’s my question, why don’t more people say the following? The standard natural deduction (I mean with intro./elim. rules and as much of harmony/normalization/conservativity/et.al. as is possible/desirable) formulation of second order logic is logic and that the incompleteness of this framework shows that the full consequence relation is mathematics.
      I mean, deduction privelgers have a very nice proof theoretic account of logicality with respect to first order logic (it deals with the tonk problematics fine). I don’t see why they can’t say that second order deduction (if properly presented so as to deal with second order tonk, plonk, quonk, etc.) is logic.
      My hypothesis is that the only reason people aren’t saying this (if they aren’t) is because the deduction privilegers have tended to be logicists about math. This is why Dummett invokes the open textured notion of proof in his Goedel article, so he can still have mathematical truth be some form of provability (incidentally, Brandom needs something similar).
      I think this is pretty unfortunate, not least because it is part of the reason that standard textbooks are so horrible with respect to proof theory now. If your notion of provability is just that it’s “open textured” then what is there to say? Is there really any epistemic gain over the platonist. You are just trading a bad metaphysical infinity for a bad epistemic one (an unaxiomatizable proof system, which is dangerously oxymoronic). But if instead you just looked at Goedel as a disproof of logicism, it would be much easier to focus on elucidating the logical (i.e. deductive) core of descriptively richer frameworks and then considering the relation between this logical core and the pure mathematics expressible in the same languages.
      Anyhow, back to reading your posts.
      I’m hoping Lance has time to engage, because he’s doing such interesting work with a mathematician that is in this neighborhood.

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    2. Catarina Dutilh Novaes Avatar
      Catarina Dutilh Novaes

      Hi Jon, sorry for the belatec reply! In fact, this is a non-reply, as I am on holiday now and typing on my phone… But I will get back to you as soon as I’m back.

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