A. Wolf wrote:
>> 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.

I'm not sure I understand this.  "True" and "false" are just arbitrary 
attributes of propositions in logic.  I read you last sentence above as saying: 
Given premises, which I assume "true", then any inference from them using 
first-order logic will be "true".  But that just means I will not be able to 
infer a contradiction (="false").  In other words, first-order logic is 

Of course if I start with contradictory premises I will be able construct a 
proof in first order logic that proves "X and not-X" which 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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