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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Received on Thu Jun 11 2009 - 16:07:09 PDT