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
consistent.
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".
Brent
>
> 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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