|From:||Chris Menzel <chris.menzel@xxxxxxxxx>|
|Date:||Sun, 30 Sep 2012 09:56:37 -0500|
On Sun, Sep 30, 2012 at 8:51 AM, William Frank <williamf.frank@xxxxxxxxx> wrote:|
Sorry, couldn't parse that.
I did indeed; it didn't even occur to me that you had second-order arithmetic in mind. You are of course correct that, in second-order Peano Arithmetic (PA), addition and multiplication are definable in terms of successor.
It would have been clarifying for you to add the crucial "second-order" qualification; for me (and I'm not alone) the meaning of "arithmetic" defaults to "first-order PA". I was further misled by your claim that "most texts define all of [the usual arithmetic operators] from sucessor", as that is not the case in any of the standard logic texts I know of (Enderton, Mendelson, Shoenfield, Boolos & Jeffrey, etc), all of which focus on first-order logic and first-order axiomatizations of arithmetic, as first-order systems of arithmetic are the only ones that are relevant to establishing Gödel's theorem and its consequences. Even dedicated texts like Simpson's _Subsystems of Second Order Arithmetic_ take all three operators as primitive because many of the most interesting second-order systems of arithmetic don't presuppose full second-order semantics — understandably, as there is no complete proof theory for full second-order validity. You have to go to a text like Manzano's explicitly devoted to logics that extend FOL to find discussions of full second-order PA. Indeed, Manzano's is only text I know of (though I'm sure there are a few others) that explicitly discusses the definability of addition and multiplication in second-order PA.
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