'When Are Relations Neither True Nor False?
in [Math]
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From: Daryl McCullough on 31 Mar 2010 07:27 Newberry says... > >On Mar 30, 3:30=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> What do you mean, no? You are proposing to equate truth and provability. >> Some sentences of the form "the Diophantine equation D(x1, ..., xn) = 0 >> has no solutions" are neither provable nor refutable. > >In which theory? It doesn't matter which theory. If it is any consistent theory in the language of arithmetic such that every true closed sentence of the form t1 = t2 (where t1 and t2 are terms built up out of 0, 1, +, *) is provable, then there are undecidable Diophantine equations. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 31 Mar 2010 08:12 Newberry says... >Why are you stating this so categorically? Look at this excerpt from >Gaifman: > >QUOTE >Line 1: The sentence on line 1 is not true. >Line 2: The sentence on line 1 is not true. > >The standard evaluation rule for a sentence of the form "The sentence >written in/on ... is true" is roughly this: >(*) Go to ... and evaluate the sentence written there. If that >sentence is true, so is "The sentence written in ... is true" , else >the latter is false. In general, any *procedure* used to evaluate the truth of sentences in a self-referential language is incomplete, in the sense that there are true sentences that are not evaluated as true by the procedure. This is easy to see: Let P be some procedure to evaluate the truth of sentences. Then consider the sentence "When procedure P is applied to this sentence, the result is not true" Contrary to your claims about sentences that fail to express a possible state of the world, the above sentence makes a perfectly definite claim: That a certain procedure applied to a certain sentence does not produce a certain result. If the procedure is actually applied to that sentence, and run to completion, then we can just check to see what the result is. What this shows is that for any sound procedure that purports to evaluate the truth of sentences, there is a true (in the sense of corresponding to the facts) sentence that the procedure fails to evaluate as true. So *every* procedure for evaluating truth of sentences is incomplete. A procedure can be complete for a certain *class* of sentences, for example, those expressible in a limited language, but there will always be an extended language that the procedure fails on. -- Daryl McCullough Ithaca, NY
From: Jesse F. Hughes on 31 Mar 2010 08:23 Newberry <newberryxy(a)gmail.com> writes: > On Mar 30, 6:07 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Newberry <newberr...(a)gmail.com> writes: >> > On Mar 28, 9:01 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> >> Newberry <newberr...(a)gmail.com> writes: >> >> > On Mar 28, 5:54 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> >> >> Newberry <newberr...(a)gmail.com> writes: >> >> >> >> > But I can. In a system with gaps Tarski's theorem does not apply. We >> >> >> > can then simply equate truth with provability. >> >> >> >> Your second sentence does not follow. You have to show that you have >> >> >> a logic in which provability turns out to be equivalent to truth. >> >> >> Tarski's theorem may not preclude this possibility, but it doesn't >> >> >> follow that you can then "simply equate truth with provability." >> >> >> > Did I say it follows? I meant that it is possible. In classical logic >> >> > withuot gaps it is impossible. Why did you not interpret what I said >> >> > this way? >> >> >> "We can then simply equate truth with provability." >> >> > It does automatically folow but we can nevertheless do that. >> >> You have to *show* that this can be done in your system. > > And to a reasonable degree I have shown it. And I do NOT mean by > pointing out that Tarski does not apply. You have not even given the rules of deduction, so how on earth have you shown "to a reasonable degree" that provability and truth are the same? (For that matter, you have not given the semantics all that clearly, but instead mentioned various statements that are neither true nor false.) -- Jesse F. Hughes "I think the burden is on those people who think he didn't have weapons of mass destruction to tell the world where they are." -- White House spokesman Ari Fleischer
From: Daryl McCullough on 31 Mar 2010 08:39 Jesse F. Hughes says... >You have not even given the rules of deduction, so how on earth have >you shown "to a reasonable degree" that provability and truth are the >same? The only possible argument for truth and provability being the same is to claim that the Peano axioms exhaust everything there is to say about arithmetic; anything that's not provable from them is just not meaningfully true or false. I've never seen anyone try to make such an argument. -- Daryl McCullough Ithaca, NY
From: Jesse F. Hughes on 31 Mar 2010 08:57
Newberry <newberryxy(a)gmail.com> writes: > On Mar 30, 6:04 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> You seem to have misrepresented Aatu's claims. Moreover, you're just >> wrong. I've argued repeatedly that some sentences of the form >> >> ~(Ex)(P & Q) >> >> occur in ordinary mathematical reasoning (and hence are useful), even >> when (Ex)P is false. An example occurred in sci.math recently. >> >> Simon C. Roberts gave a purported proof of FLT[1], by arguing: >> >> ~(En)(Ea,b,c)(a^n + b^n = c^n & a, b, c are pairwise coprime). >> >> Let's denote a^n + b^n = c^n by P(a,b,c,n) and a, b, c are pairwise >> coprime by Q(a,b,c), so that Simon's argument attempts to show that >> >> c >> >> Of course, I am *not* claiming that he proved what he claims. That's >> beside my point. A poster named bill replied that (1) is not Fermat's >> last theorem[2], which has the form >> >> ~(En)(Ea,b,c) P(a,b,c,n). (2) >> >> Arturo responded[3] by proving >> >> (En)(Ea,b,c) P(a,b,c,n) -> (En)(Ea,b,c)( P(a,b,c,n) & Q(a,b,c) ). (3) >> >> Hence, a proof of (1) yields a proof of (2) by modus tollens. > > How about this? > > (En)(Ea,b,c) P(a,b,c,n). Assumption > (En)(Ea,b,c) P(a,b,c,n) -> (En)(Ea,b,c)( P(a,b,c,n) & Q(a,b,c) ). > (3) > ~(En)(Ea,b,c) P(a,b,c,n). Modus Tollens > ~(En)(Ea,b,c) P(a,b,c,n). RAA > How about it? That is certainly valid reasoning classically. I have no idea whether it's valid in your proposed system, since you haven't said. But that is certainly not the argument that was offered in the other thread. The statement (En)(Ea,b,c) P(a,b,c,n) -> (En)(Ea,b,c)( P(a,b,c,n) & Q(a,b,c) ) was proved independently of your assumption, and no one balked. Thus, my question remains: was the argument I gave invalid? A second question comes to mind: what happened to your imagination test? You've said that (because FLT is true) you cannot picture (E a,b,c,n)( P(a,b,c,n) & Q(a,b,c) ). Yet, once you assume (contrary to fact) that FLT is false, you *can* picture it? You can't picture a green round triangle, right? What if I say: assume a round triangle exists. Can you picture a green round triangle *then*?[1] That is, does the act of making an assumption change your capacities for imagining stuff? If not, then your argument above doesn't work. If so, well, then your powers of imagination are different than mine. Footnotes: [1] Green round triangles puzzle me. As far as I can tell, you find the statement "No green triangles are round" meaningful, but the statement "No round triangles are green" meaningless. -- "[I]n mathematics there are two types of integers: primes and composites. [...] It's like how in the world there are mostly two kinds of people: male and female [...] and lots of reasons for interest in the differences." -- JSH on math/biology |