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