To:  "[ontologforum]" <ontologforum@xxxxxxxxxxxxxxxx> 

From:  Christopher Menzel <cmenzel@xxxxxxxx> 
Date:  Thu, 1 Mar 2012 20:00:40 +0100 
Messageid:  <CAO_JD6Mm4Oq2ij8CGS5CcCDFt3zGy=3+GXgNXA5gHWx0uDDHA@xxxxxxxxxxxxxx> 
On Thu, Mar 1, 2012 at 5:50 PM, William Frank <williamf.frank@xxxxxxxxx> wrote: ...a basic feature of any defined term is the principle of substitutability: you can always replace the defined term with its definition. Yes, of course, but that's where the miscommunication is here.
Here is how it works, for addition: Very important to note that you are in a framework powerful enough to quantify over functions and prove the existence of recursive functions. This was missing from your original post.
Nonetheless it is entirely true in the context I set. The context you set in your original note was "an axiomatic arithmetic for the natural numbers" which I (apparently hastily) took to mean (firstorder) Peano Arithmetic (PA) — which does not quantify over functions and, in particular, is not capable of proving your theorem above. I was explicit about this in the more formal exposition of my point, which I put below my more informal message. And, in the context of PA, it is indeed quite impossible to define addition in terms of zero and successor; if you take the usual recursive axioms for "+" to be a definition, the purported definition is neither eliminable nor conservative. However, as I'd intended the more informal part of my message to be selfcontained, it was an expository error on my part not to make the context of my assertion more explicit when I made it.
Chris Menzel _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontologforum/ Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontologforum/ Unsubscribe: mailto:ontologforumleave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To join: http://ontolog.cim3.net/cgibin/wiki.pl?WikiHomePage#nid1J (01) 
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