On 11 Jun 2009, at 14:48, 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.

I am indeed not using the term "soundness" like it is used in  
"soundness and completeness" theorem, like for first order predicate  
I use it, like many "provability logician" to mean "true in the  
standard (usual) model of arithmetic.

There, they call arithmetic soundness what me (and many logician) call  
"soundness", when they refer to theories about numbers.
Like Mendelson I prefer to use the term logically valid, to what you  
call soundness.

Should not be a problem in this list, given that we don't use the  
notion of models, nor of logical validity. I refer very rarely to  
Gödel's completness, and when I do so, I do it in the form " a theory  
has a model iff it is consistent" (this can be proved to be the case  
for first order theory).

> 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 am not sure I follow you. You mean by "true", I guess "true in, or  
satisfied by, all models", or "false in any models". A theory is sound  
if what is provable in the theory is satisfied by (true in) all models  
of the theory.
A deduction A => B is sound, or logically valid, if all models which  
satisfy  A satisfy B.

The word "true" alone has no meaning. It refers always to a model, or  
to a collection of models.

> I think what you're bickering over isn't the soundness of the system.

It is the arithmetical soundness.

>  I
> think it's the selection of the label "natural number", which is a
> completely arbitrary label.

Nooo...., come on.

> 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".

I don't see what you mean here. Robinson Arithmetic, which is a finite  
theory, can be see as a definition of the usual natural numbers, but  
like any definitions, finite or infinite (but then recursively  
axiomatisable), it has non standard models satisfying the definition.

Oh, you mean a definition of natural number such that the model would  
be finite in scope. This is non sense for me. Pace Torgny.

> 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.

Well, there is just no categorical first order definition of the  
finite sets of natural numbers. And second order definition, assumes  
the notion of infinite set.



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