Re: [ontolog-forum] Universal Basic Semantic Structures

[William Frank <williamf.frank@xxxxxxxxx>]

On Sun, Sep 30, 2012 at 10:56 AM, Chris Menzel <chris.menzel@xxxxxxxxx> wrote:

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

...

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".

That is right, you are **very far** from alone, and having lots of respectable company is something that most scientists take a great deal of comfort in, for good reasons.   This is my underlying point:
 

as first-order systems of arithmetic are the only ones that are relevant to establishing Gödel's theorem and its consequences.  

 
Yes, but the implicit computational cultural bias is that as a 'consequence' (in another sense, using an unexplicit assumption), first order arthmetic better expresses arithmetic concepts.  I think it shows the opposite, that first order arithmetic cannot adequately express arithmetic concepts.
 

 — understandably, as there is no complete proof theory for full second-order validity. 

Yes, and this "understandably" is to me the mechanistically-oriented bias of the generally accepted views.   I this attitude is behind the inadequacy of OWL, for just one example.  Why not the opposite conclusion, that people's imagination exceeds their grasp, so let us focus on our imagination, which can be expressed better in higher order languages?

So, when it comes to this thread's topic of "Universal Basic Semantic Structures", this is one of its facets, which has raised its head in several guises: the status of higher order concepts. 

Wm


 

-chris

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-- 
William Frank

413/376-8167

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