> 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. Anna --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to email@example.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---