In a Consistent System, Can a True Sentence Imply a False One?
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From: Aatu Koskensilta on 5 Aug 2010 12:04 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Secondly, not all mathematical truths about the natural numbers are > definable and hence in general if a particular formula's truth value > hasn't already been known, there's the possibility it couldn't be > unknown - in the impossibility sense. What does it mean for a truth or claim about naturals to be "definable"? The notion of the truth of a formula being impossible to know in some absolute sense is not unproblematic, and requires elucidation. Further, unless truth of arithmetical statements is connected, through some argument, explanation, theory, to our inability or ability, in some idealised sense, which again must be spelled out, to know such truths, nothing whatever follows about the relativity of such statements from our possibly necessary inability in some instances to know their truth. > In a sense, Tarski's Undefinability of Truth has already established > the concept of the naturals as a relativistic concept, with > "undefinable" being an alias, if not a codeword, for > "unknown-ability", hence "relativity". There is no absolute notion of definability (or of undefinability) in logic. Tarski's theorem about the undefinability of truth doesn't imply that arithmetical truth is in any absolute sense undefinable, only that it is not arithmetically definable. Indeed, we can give a perfectly explicit definition of arithmetical truth using standard machinery of ordinary mathematics -- of similar level of abstraction as we meet in e.g. graph theory, results about embeddability of finite trees, etc. > On the other hand, from the practicality point of view, of mathematics > being a language of physics and sciences, given that the real and complex > numbers are definable from the naturals with basic concepts of set > operations (e.g. equivalence classes), it'd seem worthwhile to try to > come up with criteria to determine what would be absolute truths and > what would be relative ones. A necessary first step to this line of investigation is to clarify for ourselves just what sort of absoluteness and relativity is in question. We don't need -- and indeed can't in general expect -- any formal explication, but we need a clear informal explanation, detailed and precise enough that we can be sure we know what we're talking about. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Daryl McCullough on 5 Aug 2010 14:17 In article <BOE5o.47972$xZ2.27206(a)newsfe07.iad>, Nam Nguyen says... >Secondly, not all mathematical truths about the natural numbers are >definable and hence in general if a particular formula's truth value >hasn't already been known, there's the possibility it couldn't be >unknown - in the impossibility sense. First of all, it's not the case that the truths of arithmetic are undefinable. It's just that they cannot be defined using a formula of arithmetic. You can certainly define the truths of arithmetic using very weak set theory. Second, it is hard to know how to make sense of the concept of a statement being *impossible* to know. For any particular formal system, there are sentences that cannot be proved in that system, but those sentences can be proved in some other formal system. I suppose what you might mean is that there might be a sentence of arithmetic that can never be proved in any formal system with "self-evident" axioms. There is a sense in which the axioms of PA are close to being self-evidence, since they just formalize what we *mean* by the natural numbers, the number 0, and the operations of plus, times, and successor. Reflection allows us to come up with new theorems that are *almost* as self-evident. Since PA just captures manifestly true facts about the naturals, it follows that PA is consistent, and so the formalized statement Con(PA) is almost self-evidently true. I'm not sure if there is a formal characterization of the self-evident statements of arithmetic. I think not, because if we had a formal characterization, then we could formulate the corresponding Godel sentence: G <-> G is not self-evidently true Then G would have to be true. And sense we can clearly see that it has to be true, it would be self-evidently true, which is a contradiction. So I think that "self-evidently true" must always be an informal notion. >In a sense, Tarski's Undefinability of Truth has already >established the concept of the naturals as a relativistic >concept, No, it doesn't do that. Tarski didn't show that arithmetic truth is undefinable, just that it can't be defined by a formula in the language of arithmetic. The truth of first-order arithmetic can be defined in set theory, or in second-order arithmetic, for example. We can actually come close to defining truth in the language of arithmetic. For each natural number n, there is a formula of arithmetic True_n(x) such that True_n(x) <-> x is the code of a true formula of arithmetic that uses n or fewer quantifiers. -- Daryl McCullough Ithaca, NY
From: George Greene on 6 Aug 2010 07:05 On Aug 4, 5:04 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > A couple of the other arguments that happen to stand out in my mind > were: > > (1) He wouldn't agree that there is such a thing as the theory of a > model (the set of sentences true in a given model), though it was > explained to him over a dozen times and is found in virtually any > textbook in mathematical logic. Well, I don't agree with that either. Syntax is one thing. Semantics is another. The theory is one thing. The model is another. When I first brought this literally umpteen years ago, the smackdown was "Oh, YOU mean a FORMAL theory". The point is that the theory is inherently investigative. It is an attempt TO CHARACTERIZE the set of truths of the model IN A FINITE way. What you are calling "the theory of the model" is in fact a "theory" where EVERY truth of the model IS AN AXIOM. This then becomes analogous to DMC's arguments about the irrelevance of the definitional difficulties that arise in vacuous cases, because this case IS VACUOUS: the PURPOSE of "having a theory" of the model is to have some finitarily describable (I.E. RECURSIVE) set of axioms that will suffice as a finitary "abbreviation" for (by implying the remaining whole) the infinitary (i.e. r.e. but NOT totally recursive) set of truths of the model. The whole implication of Godel's 1st incompleteness theorem is that in the case of "true" first- order arithmetic (even degenerate forms of it like Robinson arithmetic), NO SUCH AXIOMATIZATION IS POSSIBLE. Therefore, talk about theories in this context IS STUPID. The field's standard parlance is arguably just WRONGLY overloading the term. They are defining "theory" too broadly and then forcing a longer modified term ("formal theory") into service for the MORE fundamentally RELEVANT concept. > (2) The whole thing about deriving (in a language that has '0' as a > constant symbol) 0=0 from the sole axiom Ax x=x (or whatever the exact > example was). The more general matter that a formula with symbols that > do not occur in a given set of axioms may still be derivable from that > set of axioms. No, they are not, not if the theory is consistent. GOOD GRIEF. THE LANGUAGE COMES *FIRST*!! JEEzus.
From: George Greene on 6 Aug 2010 07:08 On Aug 4, 5:04 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > (2) The more general matter that a formula with symbols that > do not occur in a given set of axioms may still be derivable from that > set of axioms. NO, you do NOT get to derive Pv~P in a language that does not have P in it. You FIX the language IN ADVANCE and the ONLY question being investigated is what theorems follow from the axioms. If the language has symbols that do not occur in the axioms, then, again, you are dealing WITH A VACUOUS case. There is simply nothing to be gained by having those symbols in the language. A lot of this is arising out of cases where there is some intended model or language in advance. That is NOT appropriate EITHER. The AXIOMS come first and LOGIC IS ABOUT what follows from the axioms and/or the rules of inference.
From: Daryl McCullough on 6 Aug 2010 10:34
George Greene says... > >On Aug 4, 5:04=A0pm, MoeBlee <jazzm...(a)hotmail.com> wrote: >> A couple of the other arguments that happen to stand out in my mind >> were: >> >> (1) He wouldn't agree that there is such a thing as the theory of a >> model (the set of sentences true in a given model), though it was >> explained to him over a dozen times and is found in virtually any >> textbook in mathematical logic. > >Well, I don't agree with that either. >Syntax is one thing. Semantics is another. >The theory is one thing. The model is another. >When I first brought this literally umpteen years ago, >the smackdown was "Oh, YOU mean a FORMAL theory". >The point is that the theory is inherently investigative. It is an >attempt TO CHARACTERIZE the set of truths of the model IN A FINITE way. The model theoretic characterization of what it means for a sentence to be true in a structure gives a finite characterization of the complete theory associated with a model. It isn't necessarily enough information to allow you to generate the sentences in the theory, but you can prove facts about those sentences. I suppose it depends on how you are defining what a "theory" is. You can define it as "A collection of sentences closed under logical deduction", in which case, there is no requirement that there be a computable way to generate the sentences. Or you can define it operationally, as the set of sentences that can be generated from a particular set of axioms. I'd rather use the more general definition, and then use a phrase such as "axiomatizable" to mean r.e. theories. -- Daryl McCullough Ithaca, NY |