> 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.

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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.
Anna
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