'When Are Relations Neither True Nor False?
in [Math]
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From: Daryl McCullough on 25 Feb 2010 11:25 Newberry says... > >On Feb 25, 4:07=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> Look, consider the theory with no axioms. Can you prove >> 0 ~= 1? No, you can't. Is it true? Of course, it is. So >> truth and provability are not the same. That's obvious. > >The question was why I was trying to eliminate the duality. And I >thought that the answer was obvious. No, it seems obvious to me that there is a distinction between truth and provability. >> As I said, we know that 0 ~= 1 prior to any notion of proof. We >> choose our axioms and our proof theory so that true statements >> are provable. > >Indeed. When we are making logical inferences we are preserving the >truth and we are able to formalize the steps syntactically. So when we >arrive at the conclusion that G is true how did we fathom it without >being able to formalize it? We prove an implication: If PA is consistent, then G. We don't have a proof (within PA) that PA is consistent, but we believe it's true. So G follows from the assumption that PA is consistent. -- Daryl McCullough Ithaca, NY
From: MoeBlee on 25 Feb 2010 11:31 On Feb 24, 11:00 pm, Newberry <newberr...(a)gmail.com> wrote: > How can you arrive at the conclusion that something is true other than > by a proof? We can use the ordinary model theoretic notion of truth, and prove a sentence to be true in a particular model but not provable in a particular theory. For example in Z set theory we prove that the Godel sentence for PA is true in the standard model for the language of PA while the Godel sentence for PA is not provable in PA. (And 'true in the standard model for the language of PA' is often spoken of as just 'true'). And even aside from such formalities, given just ordinary methods of mathematical reasoning (ingeniously applied) we may safely conclude that the Godel sentence for PA is true. I mean, if you doubt ordinary arguments that the Godel sentence for PA is true, then you might as well doubt a whole range of ordinary mathematical arguments. MoeBlee
From: MoeBlee on 25 Feb 2010 11:41 On Feb 24, 11:54 pm, Newberry <newberr...(a)gmail.com> wrote: > On Feb 24, 12:31 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) > > Godel's theorem says that for every theory T of the right type, > > there is a formula Phi such that neither Phi nor ~Phi is provable > > in T. > > This is correct. But it would be incorrect to claim that every such > Phi in every such theory must be interpreted as true. No one said otherwise. However, a great many people do accept as plainly true the sentences that are true in the standard model for the language of PA. > If the theory is sound then what it can or cannot prove will depend on > the interpretation. A theory is sound if all its members (a theory being a certain kind of set of sentences) are true. But that raises the question, "true in WHAT model?" Ordinarily, for theories in the language of PA, we take the determining model to be the standard model for the language of PA. > If there are sentences that are neither true nor > false then it will not be able to (dis)prove them. There ARE no sentences that are neither true nor false in a given model. Given a model for a language, the function that maps sentences to {true false} is a function that is total on the entire domain of sentences of the language. If P is a sentence of the language and M is a model for the language, then the truth function for M assigns to P the value 'truth' or assigns to P the value 'false'. (In all of this I'm referring to ordinary classical mathematical logic.) MoeBlee
From: MoeBlee on 25 Feb 2010 11:43 On Feb 25, 12:00 am, Newberry <newberr...(a)gmail.com> wrote: > On Feb 24, 9:17 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > Newberry <newberr...(a)gmail.com> writes: > > > Anyway, whethere GIT suggests that two valued logic is impossible or > > > not is not essential to my main point. > > > Well, I don't really have any interest in your main point. I was just > > wondering why the incompleteness theorem suggests to you that "two > > valued logic is impossible". Apparently it's impossible to say. > > I do not know why you have any interest in knowing why GIT suggests > that bi-valent logic is impossible, considering that you have already > made up your mind. Whether he's made up his mind on certain matters or not, it is not precluded that he may be interested in why you say that the incompleteness theorem suggests that two-valued logic is impossible. I too am interested in why you say that the incompleteness theorem suggests that two-valued logic is impossible. MoeBlee
From: MoeBlee on 25 Feb 2010 11:49
On Feb 25, 8:52 am, Newberry <newberr...(a)gmail.com> wrote: > When we are making logical inferences we are preserving the > truth and we are able to formalize the steps syntactically. So when we > arrive at the conclusion that G is true how did we fathom it without > being able to formalize it? If you demand formalization, then we can formalize in a meta-theory such as Z set theory. That is, e.g., Z set theory proves that the Godel sentence for PA is true in the standard model for the language of PA, as we also may formally prove (and we can do it in even weaker meta-theories than Z set theory) that the the Godel sentence for PA is not a theorem of PA. MoeBlee |