Re: [ontolog-forum] Universal Basic Semantic Structures

[Chris Menzel <chris.menzel@xxxxxxxxx>]

On Sun, Sep 30, 2012 at 8:51 AM, William Frank <williamf.frank@xxxxxxxxx> wrote:

...

arithmetic has the operations plus and minus and times and successor.  Of course, most texts define all of them from sucessor. 

You are quite mistaken.

I hope I would be as unlikely to be '*quite* mistaken' as if someone would make clear assertions about operations in set theory and yet not know that subset could be defined in terms of membership.  

Sorry, couldn't parse that.

 

It is impossible to define addition in terms of successor and multiplication in terms of addition and successor. You appear to be mistaking the usual recursive axioms for those operators for definitions. They are not. Full arithmetic requires all three operations. (Interestingly, exponentiation can be defined in terms of addition and multiplication.)

I think what you meant was,  'in a first order axiomatization of arithmetic'.

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.

-chris

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