Torkel Franzen on truth
in [Logic]
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From: MoeBlee on 8 Jan 2008 13:50 On Jan 7, 3:22 pm, herbzet <herb...(a)gmail.com> wrote: > MoeBlee wrote: > > To express what you just mentioned, we also have <A T> where A is an > > axiomatization of T. > > Does "axiomatization of T" mean "recursive axiomatization of T"? > Or can A be any old set of sentences? I think authors differ. As best I recall having read various authors, by 'axiomatization', some mean any set of sentences that entails T, while others mean a recursive set of sentences that entails T. Personally, I go with the first sense and use 'recursive axiomatization' for the second sense. MoeBlee
From: george on 8 Jan 2008 14:52 > > Does "axiomatization of T" mean "recursive axiomatization of T"? > > Or can A be any old set of sentences? > > I think authors differ. It doesn't matter. Why you are so obsessed with what "authors" do is utterly beyond me. You can do anything you want as long as it is consistent. If the authors weren't smart enough to do likewise then that is their problem, not yours. > As best I recall having read various authors, > by 'axiomatization', some mean any set of sentences that entails T, > while others mean a recursive set of sentences that entails T. "As best I can recall" is simply neither adequate nor relevant. The issue is any case is not whether the AXIOMS are vs. aren't "any" set of sentences, BUT RATHER, whether THE THEORY is vs. isn't "any" set of sentences. There are 2 kinds of theories in the world, namely those that are r.e. and those that aren't. THAT is what matters. Just because that wasn't the question the questioner asked does NOT matter. The questioner doesn't always KNOW what matters. You should, though. > Personally, I go with the first sense and use 'recursive > axiomatization' for the second sense. You need some help from the dictionary. The "-ization" suffixes connote a process, a method, for actually finding or providing axioms in the theory. If the theory is not r.e. then this is basically not possible except in one of your usual trivial degenerate senses (like the one in which every structure is a model). The correct answer to the question is that if the theory is r.e. then it will have a recursive axiomatization, and if the theory is not r.e. then you have no hope of ever telling what a theorem is, let alone what an axiom is, so the alleged theory is hardly even worthy of the name. In other words, all axiom-sets that can reasonably or productively be thought of as axiom-sets are necessarily recursive. If the axiom-set is more complicated than r.e. to begin with, then obviously the usual rules of standard classical FOL are already in over their head.
From: MoeBlee on 8 Jan 2008 16:40 On Jan 8, 11:52 am, george <gree...(a)cs.unc.edu> wrote: > > > Does "axiomatization of T" mean "recursive axiomatization of T"? > > > Or can A be any old set of sentences? > > > I think authors differ. > > It doesn't matter. > Why you are so obsessed with what "authors" do is utterly > beyond me. I'm not obsessed with authors. What a silly remark by you. It's just that this subject is mainly disseminated through lectures and writings, and primarily, for those who are not gathered in a single lecture series, widely used textbooks provide the most common basis for definitions. I was asked what a term "means". I take that to be a request for what the term means to the people who use it as a technical term in the subject. So my best answer is to say what I know about how various authors use the term. > You can do anything you want as long as it > is consistent. If the authors weren't smart enough to do likewise > then that is their problem, not yours. Yes, I've said so myself that one can set up one's own system, and with one's own system of definitions, either keeping to the ordinary senses or departing from them. Usually, if one departs from ordinary senses, then I think one should have some reason for doing that. Indeed, authors differ among themselves, as they find convenient for the purposes of their own treatment. And I have culled from different books, and put together, in my typed notes, my own treatment (with much unfinished) with my own system of definitions, keeping to ordinary senses generally but tweaking sometimes to suit my own treatment. Moreover, do you at least see how what you're saying now about "you can do anything you want as long as consistent" goes against the grain of your continual harping ('harping' is putting nicely) that people are NOT within intellectual prerogative to go against the "paradigm" (as you've called it) and the presumed definitions? If I take you up now on your "do anything you want as long as consistent" and define, in some consistent system of definitions, a term so that it makes good sense but goes against the "paradigm", then I can pretty much count on you berating me for arrogating to myself a prerogative to go against the presumed definitions. Anyway, as I said, the poster asked what the term "means", so I take the context not to be what I stipulate the term means but what it means as ordinarily sed in the field of study. (And I did go on to say which meaning I adopt in my own terminology.) > > As best I recall having read various authors, > > by 'axiomatization', some mean any set of sentences that entails T, > > while others mean a recursive set of sentences that entails T. > > "As best I can recall" is simply neither adequate nor relevant. Oh, for godsakes. It's just a minor and casual disclaimer. I have never meant anyone as thoroughly and gratiutously disputatious as you. Do you have a disorder or something that causes you to need or thrive on that? > The issue is any case is not whether the AXIOMS are vs. aren't > "any" set of sentences, BUT RATHER, whether THE THEORY > is vs. isn't "any" set of sentences. You say, "THE issue" [emphasis added]. You do that, and similar, often. Maybe in this particular case you don't mean to be saying that the issue you wish to address makes the issue other people are addressing a distraction or aside from what is important, or similar, but it at least seems that way, and in certain other instances it has been that way. People like to discuss various aspects of these subjects. We don't need you to harp as to what "THE issue is". The poster asked me about a certain term, and I answered him. To do that, I don't need to check as to what George considers "THE issue" in a given discussion. > There are 2 kinds of theories > in the world, namely those that are r.e. and those that aren't. > THAT is what matters. And in certain other contexts people may wish to consider whether a certain set of sentences, - whether recursive or not - is a set that proves some other set of sentences. Please, give it a break already! > Just because that wasn't the question the questioner asked > does NOT matter. The questioner doesn't always KNOW what > matters. You should, though. PLEASE! I don't post to fully screen each question and remark as to what is THE matter of greatest import! That would be an absurd obligation to demand of anyone. There are lots of different items of discussion that get into a subject, ranging from trivial to small to important to, by SOMEONE's (YOURS?) ordering of importance. I, as do other reasonable people, feel well within bounds of reasonablity by fielding various items within that range as suits both my intellectual and recreational purposes for posting and without submitting to whatever hierarchy of importance YOU have decided upon at each and every context in an ever changing stream of contexts. > > Personally, I go with the first sense and use 'recursive > > axiomatization' for the second sense. > > You need some help from the dictionary. The "-ization" suffixes > connote a process, a method, for actually finding or providing > axioms in the theory. Again, how utterly captious of you. Mathematical terms don't always conform perfectly to dictionary or English grammar constraints. The word 'axiomatization' in mathematics is well established, and is clear in the sense I use it, and I can formalize it to any degree of formality required. > If the theory is not r.e. then this is > basically > not possible except in one of your usual trivial degenerate senses > (like the one in which every structure is a model). A theory T (as a set of sentences in a language such that the set is closed under entailment) is recursively axiomatized iff there exists a recursive set of sentences S in the language such that T is the set of consequences of S. That is a perfectly acceptable (and common) definition. > The correct answer to the question is that if the theory is r.e. > then it will have a recursive axiomatization, I correctly answered the question asked. And, as you said, if a theory is recursively enumerable then it has a recursive axiomatization. I have no issue with that. > and if the theory is > not r.e. then you have no hope of ever telling what a theorem is, > let alone what an axiom is, so the alleged theory is hardly even > worthy of the name. If the theory is not recursively enumerable then membership in the theory is not decidable. But that doesn't preclude discovery that certain sentences are or are not members of the theory. As to being "worthy" of being called a theory, we do understand that a theory that is not recursively enumerable does not fit certain informal notions of what a theory is or even other technical definitions of 'theory'. We don't promise that every technical meaning of a word such as 'theory' is faithful to every informal sense also or to other contrasting technical definitions of 'theory'. > In other words, all axiom-sets that can reasonably or productively > be thought of as axiom-sets are necessarily recursive. There is no problem in allowing a distinction between a recursive axiomatization and an axiomatization that is not recursive. > If the axiom-set is more complicated than r.e. to begin with, then > obviously the usual rules of standard classical FOL are already > in over their head. I'm not sure what you mean, but at least I do agree that for certain purposes, recursive axiomatizations are the ones we're interested in, as I have harped on that point in many a discussion with people who don't appreciate the importance of recursive axiomatization, but that doesn't preclude that it is reasonable to define 'theory' so that there are also theories that are not recursively axiomatizable, and that if you want to define 'theory' so that theory has a recursive axiomatization, then, fine, live and let live. For an extended technical discussion, we would have to agree on one definition or the other, but in a context of posting, I've given a clear and ordinary definition that serve the purpose especially as a conversation can be facilitated by agreement upon a definition that is the one used in what is arguably the most widely used and referenced textbook in the subject (or at least among the most widely used and referenced), and not with the argument that it is a good definition simply for being the one used in that textbook but rather that at least that textbook provides a common reference even as we may agree to depart from it or qualify it on certain definitions as suits our purpose. MoeBlee
From: herbzet on 9 Jan 2008 01:02 george wrote: > MoeBlee wrote: > > herbzet wrote: > > > Does "axiomatization of T" mean "recursive axiomatization of T"? > > > Or can A be any old set of sentences? > > > > I think authors differ. [...] > > As best I recall having read various authors, > > by 'axiomatization', some mean any set of sentences that entails T, > > while others mean a recursive set of sentences that entails T. [...] > Just because that wasn't the question the questioner asked > does NOT matter. The questioner doesn't always KNOW what > matters. The point of my question, such as it was, is that the first sense of the definition reduces to Enderton's definition. -- hz
From: george on 9 Jan 2008 13:14
On Jan 8, 4:40 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > Moreover, do you at least see how what you're saying now about "you > can do anything you want as long as consistent" goes against the grain > of your continual harping ('harping' is putting nicely) that people > are NOT within intellectual prerogative to go against the > "paradigm" (as you've called it) and the presumed definitions? You have this exactly backwards. *I* am going against the presumed definition in this case. |