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  
logic.
I use it, like many "provability logician" to mean "true in the  
standard (usual) model of arithmetic.

See
http://en.wikipedia.org/wiki/Soundness
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.

Bruno


http://iridia.ulb.ac.be/~marchal/




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