> On 18 Dec 2018, at 23:14, Brent Meeker <[email protected]> wrote: > > > > On 12/18/2018 6:20 AM, Bruno Marchal wrote: >>>> By definition; soundness means that it reflect reality. >>> >>> You're now messing with words. What does "reflect reality" mean? It looks >>> like an appeal to correspondence theory to truth. But that means to know a >>> theory is sound you need to know what is true. >> >> >> There are two definition of soundness. >> >> Arithmetical soundness: it means that the theorem are true in (N, 0, +, *) >> >> Exemple if the theory proves ExEyEx(x^3 + y^3 + z^3), if the theory is >> sound, it means that there are (standard) natural numbers n, m, r such that >> their sum of cube gives 33. >> >> Soundness (in general): it means that what is proved in the theory is true >> in all models/intepretations of the theory. >> >> Completeness is the reverse of that one; a theory is complete if what is >> true in all models is provable. > > That's ambiguous. Do you mean a theory is complete if whatever is true in > some model is provable, or do you mean that whatever is true in every model > is provable?
That last one. In first order logic: []p is equivalent with [p is satisfied (is true) in all models] <>t is equivalent with [there exists a model where p is true] (Like in the modal logic, where []p means p is true in all accessible worlds, and <>t means that there is at least one accessible world). (Note that in a cul-de-sac world, all []# are “trivially” satisfied. > Since there are in general arbitrarily many models, is completeness only > knowable by a proof in a metatheory? Yes. Completeness of a theory can be proved only in some meta theory, but with the Löbian complexity threshold we can get theories able to prove a large part of their metatheory, sometimes conditionally, like If I am consistent, then I cannot be complete. PA proves that. Technically there is a sense to say that PA, or sometimes PA + another axioms, can prove their own metamathematics. The Löbian machine proves their own second Gödel’s incompleteness theorem, and Löb’s theorem (they are Löbian!), but can also prove their own Tarski theorem, but that is more difficult to shown and counter-intuitive as no theory can prove the existence of a model for itself. Bruno > >> >> Not to confuse with Arithmetical Completeness: a theory is arithmetically >> complete if it proves all truth in (N, +, *). >> >> PA is a complete theory, like all first order theories (Gödel 1930) >> PA is arithmetically incomplete (Gödel 1931). >> >> You can prove that PA is sound using a bit of set induction, like in >> Analysis. Humans tend to trust Analysis, which assumes much more than >> arithmetic. >> >> >> >> >> >> >>> >>> Sound just means its theorems are tautologies, i.e. they are valid >>> inferences from the axioms. >> >> You might makes number theorists a bit nervous if you say that Fermat is a >> tautology. We use just “theorem”, and use only tautology for classical >> propositional logic. > > But they are conceptually the same. > > Brent > >> >> Sound means true in all models (arithmetically sound means true in the >> standard model (N, +, *). >> >> >>> >>>> >>>> Soundness implies consistency. But consistency does not imply soundness. >>>> The robot describing the Venus of Milo in front of another sculpture is >>>> consistent, but unsound. >>>> >>>> All the machines I am talking about are supposed to be arithmetically >>>> sound. >>> >>> Meaning that what they "believe" are theorems. >> >> No, meaning that the theorem/beliefs are true in the structure (N, +, *). It >> means that if PA would prove ExEyEx(x^3 + y^3 + z^3), then there are really >> numbers n, m, r, having their cube added to 33. With PA + []f, you might be >> able to prove ExEyEx(x^3 + y^3 + z^3), but still not know if such standard >> n, m, and r exist, because x, y and z could be non standard natural numbers. >> PA + []f is typically unsound. It asserts that there is proof of the false, >> but can prove that it not 0, nor 1, nor 2, nor 3, nor 4, etc. It is >> consistent, but not omega-consistent. >> >> >> >> >>> Not that they "believe" everything that is true. In fact you have proven >>> anything about what a person might or might not believe. Your ideal >>> machines do not include any concept of acting on "beliefs" which is the >>> real test of beliefs. >> >> I have no clue why this could be true, except contingently, as I use all >> this to derive physics from arithmetic, not for doing machine acting in our >> local neighbourhood. >> >> Bruno >> >> >> > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at https://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
