> 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.

I may have misstated myself, but the wiki article you pointed me to agrees 
with what I tried to say: A logical system is sound if every provable 
statement is valid.  Validity is not the same as soundness.  There are valid 
arguments that are unsound.  For example, if I say "x is not equal to x, 
therefore there are no more than five natural numbers", this is a valid 
(i.e., logically true) argument.  But it's also an unsound argument, because 
there is no interpretation where x is not equal to x.

What you're calling soundness I would call omega-consistent, but I see from 
the article that this is sometimes called "arithmetical soundness".

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

One could make the same argument about the symbol "=" not having any meaning 
outside of a model, but "true" has a standard meaning in logic, one that is 
often used interchangeably with "valid" (a stronger property).  The general 
"true" means "true under any interpretation".

> 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.

Nonsense for me too, apart from the philisophical musings.

> 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.

I'm not sure what you mean here.  Of course there is no categorical 
first-order theory of N.


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