> 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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