Torkel Franzen on truth
in [Logic]
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From: george on 2 Jan 2008 11:06 On Jan 2, 11:02 am, george <gree...(a)cs.unc.edu> wrote: > For ANY theory rich enough to formulate Con(S) > or G_S at all, S doesn't prove G_S. Formal theory. I was thinking about a theory as the closure of a recursive axiom-set. That some random collection of strings that is not even r.e. even qualifies as "a theory" (at all) is an abuse of language. I am every bit as opposed to referring to a NON-formal theory as "a theory" as you are to referring to a formal language as "a language".
From: MoeBlee on 2 Jan 2008 17:18 On Dec 20 2007, 8:05 pm, G. Frege <nomail(a)invalid> wrote: > On Thu, 20 Dec 2007 19:51:17 -0800 (PST), MoeBlee <jazzm...(a)hotmail.com> > wrote: > > > Of course we know that there are senses in which a formal language is > > a set of strings, also a sense in which the language is taken not to > > be the set of strings but rather a tuple that "encodes" a certain > > signature and sets of kinds of symbols (cf., e.g. Monk's textbook, > > which, for its elegance and rigor, I think is one of the best > > definitions of 'first order language'), and other senses of language > > as purely syntactical. > > > But there are other notions (even if more in the minority these days) > > in which a language is a syntax and a semantics. And even of a > > language as a syntax, semantics, and pragmatics. And perhaps other > > conceptions of 'language' and 'formal language'. > > > So, as long as Peter is clear that he's just giving his definition for > > the purpose of conveying his exposition, I see nothing wrong in that, > > and more power to him. > > I guess this (your account) is a reasonable approach. > > george, on the other hand, originally claimed: > > "Prof. Smith and I have been talking, for example, about the > intended model vs. the formal language. He is the one who > said that he didn't think formal languages should even be referred > to as a language. That is considerably less defensible than anything > *I* have ever said." > > And _that_ (i.e. george's last claim) is certainly not reasonable > (imho). Smith gave arguments (whether I agreee with them or not) for his view that 'langauge' is not suited to refer to systems that are merely syntactical. He also allowed that it's fair enough though that the word is used stipulatively (as long as the stipulation is clearly given) in that way, though he thinks it not best. Meanwhile, it is a fact that some authors in mathematical logic do use 'language' in the way Smith prefers and it is my view that that usage too is fair enough as long as its stipulation is clearly given too. MoeBlee
From: MoeBlee on 2 Jan 2008 17:21 On Dec 20 2007, 8:10 pm, G. Frege <nomail(a)invalid> wrote: > Yes, yes, we know good old george. :-) On the other hand, his posting > manners have improved considerably in recent years. I see no reason to think that. MoeBlee
From: MoeBlee on 2 Jan 2008 17:54 On Dec 22 2007, 9:17 am, george <gree...(a)cs.unc.edu> wrote: > On Dec 20, 8:19 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > I just let go by uncontested thousands of words of yours, some of them > > HILARIOUSLY ill-conceived (I just love your 0 = {x | Ay ~yex}, which > > you posted to CORRECT a bunch of people who had ALREADY CORRECTLY > > observed various equations with 0 !!!) > > That was simply a mistake. Sure, we all make mistakes. What makes it notable though is the context of you harranging a bunch of other people who were not mistaken. > Everybody (including you) excused it as such UNTIL NOW. > You are lapsing. You had it right the first time when you just > let it go by. And there was not any "already correctly" going on in > that > context. Nam had said one thing wrong and FF had (as usual) > corrected it INcorrectly. Nam is out to lunch; we know that. But the poster Frege's remarks were correct. He didn't say his formulation is a DEFINITION of the empty set. > > and also intellectually > > hypocritical (the line you're arguing about semantics and axioms > > lately is the EXACT NEGATION of the line you argued, rather by > > spraying your mouth-foam in my face, when we first exchanged posts), > > Put up or shut up. The thread you renamed to 'tedious sledding [...]' and your crazed attack of my perfectly sensible distinctions regarding provability and semantics as opposed to your latest mood swings on that subject. > I do not go around posting undocumented lies about people. > Since you now appear to be doing exactly that, I will be content > for us to rmain enemies. Oh good, a toxic dump is declaring me an enemy. > > since I learned a while ago that not only is there no point trying to > > get through to you but doing so is an ESPECIALLY unpleasant endeavor. > > I certainly don't mind people who are more ignorant than I am > "giving up" on trying to persuade me to agree with their errors. > > I do, however, mind people lying about me or claiming that I said > X without being able to back it up. You seriously deflated the meaning of 'lying' a long time ago. MoeBlee
From: MoeBlee on 2 Jan 2008 18:41
On Dec 22 2007, 2:03 pm, "Nam D. Nguyen" <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > On Dec 19, 3:11 pm, "Nam D. Nguyen" <namducngu...(a)shaw.ca> wrote: > >> tc...(a)lsa.umich.edu wrote: > >>> In article <q8gaj.19679$Tx.4697(a)pd7urf3no>, > >>> Nam D. Nguyen <namducngu...(a)shaw.ca> wrote: > >>>> tc...(a)lsa.umich.edu wrote: > >>>>> (1) There are no nonzero integers m and n such that m^2 = 2 n^2. > >>>>> (2) In the standard model of the integers, there are no nonzero integers > >>>>> m and n such that m^2 = 2 n^2. > >>> [...] > >>>>> Does that mean that (2) is absolutely, unconditionally true? > >>>> No. What is a standard model of the integers might not be the "standard" > >>>> model to others! > >>> All right, then, let's try this one: > >>> (3) In every model of PA, there are no nonzero integers m and n such that > >>> m^2 = 2 n^2. > >>> Most people would agree with (3), since the proof of the irrationality of > >>> sqrt(2) can be formalized in PA, and therefore the statement holds in all > >>> models of PA, standard or nonstandard. > >>> Do you believe (3)? Is (3) absolutely true? Whether or not your "standard" > >>> model is the same as mine makes no difference, since the assertion holds in > >>> every model, so I don't see where relativism enters. > >> Apparently you've missed my last post where I made the correction: > > >> > Let me re-phrase it: what might be "integers" or "standard" to one, > >> > might not be to the others. > > >> Any rate, so (3) is true, *relative* to PA's models: the *truth* of (3) > >> is a relative value, and the relativity is still there! > > > (1) No, you completely dodged the point. > > No I'm not. See my latest response to TC. There you just spin more variations on your basic misunderstandings. > > Okay, in a technical sense, '2=1+1' is true relative to models because > > it's true in some models but not in others. > > In a technical sense, "2=1+1" is true in all models of some consistent theories > where it is a theorem. What you just said boils down to: '2=1+1' is true in all models of '2=1+1'. Yeah, we know that. > > But the challenge put to you was to say in what way (3) is relative to > > models, since it's not a matter of being true in some models and not > > in others, but rather of being true PERIOD, since it is a statement > > about ALL models. To reinforce that it is a statement about all > > models, please recognize that it is of the form, Given ANY model, if > > it is a model of PA, then [...]. You didn't answer that. > > (2) What is the relativity in '0011' being the same string as '0022 > > with 1 substituted for 2'? > > Quite a few ways of relativity! For example, what did *you* mean by "substituted"? > Were you referring to some kind of _mapping_? I mean it in the ordinary English sense of the word. If you point is that words can be defined differently, so that the truth of a statement is relative to how we define the words, then I don't think anyone would disagree with you. 'Meryl Streep has blonde hair' is true depending on what we mean by 'Meryl Streep', 'has', 'blonde' and 'hair'. But I don't think that is what is at issue. Given the ordinary informal English meanings of the words used in (2), don't you think it is a non-religious, objective observation that they are the same string? MoeBlee |