> As I said, you can formalize the notion of soundness in Set Theory.  But 
> this adds nothing, except that it shows that the notion of  soundness has 
> the same level of complexity that usual analytical or  topological set 
> theoretical notions. So you can also say that  "unsound" means violation 
> of our intuitive understanding of what the structure (N,+,*) consists in. 
> We cannot formalize in any "absolute way" that understanding, but we can 
> formalize it in richer theories  used everyday by mathematicians.

You're using soundness in a different sense than I'm familiar with. 
Soundness is a property of logical systems that states "in this proof 
system, provable implies true".  Godel's Completeness Theorem shows there 
exists a system of logic (first-order logic, specifically) that has this 
soundness property.  In other words, nothing for which an exact and complete 
proof in first-order logic exists, is false.

Soundness is particularly important to logicians because if a system is 
unsound, any proofs made with that system are essentially meaningless. 
There are limits to what you can do with higher-order logical systems 
because of this.

I think what you're bickering over isn't the soundness of the system.  I 
think it's the selection of the label "natural number", which is a 
completely arbitrary label.  Any definition for "natural number" which is 
finite in scope refers to a different concept than the one we mean when we 
say "natural number".  Any finite subset of N is less useful for 
mathematical proofs (and in some cases, much harder to define--not all 
subsets of N are definable in the structure {N: +, *}, after all) than the 
whole shebang, which is why we immediately prefer the infinite definition.


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