'When Are Relations Neither True Nor False?
in [Math]
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From: Nam Nguyen on 5 Apr 2010 01:08 Marshall wrote: > On Apr 4, 9:33 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Apr 2, 5:32 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> Let me put to rest the idea we know enough about the natural numbers, >>>> to prove important thing such as the consistency of PA. I'll do that >>>> by pointing out the existence of a specific unknown natural number. >>> Why do you think the existence of a specific unknown number >>> should have anything to do with consistency? >> Because they (the syntactical proof of consistency and collectively >> many formulas about this unknown natural) both connote the same thing >> in meta level: impossibility of syntactical proof. If you can't prove >> a certain formula related to this number, you can forget about proving >> a consistency, syntactically speaking. > > Both of these sentences are just a restatement of the position that > I asked you for support of. My question is not, do you consider > a proof of consistency of PA and the existence of an unknown natural > number to be mutually incompatible. You have clearly stated so. Huh? I'm not quite following you here. First of, you clearly asked me why the existence of a specific unknown natural number "should have anything to do with consistency [proof]". And I gave you an answer for that: their both connoting an "impossibility of [some] syntactical proof" would be their "having anything to do" with each other. That's all I said here. And I didn't state anything about them being "mutually incompatible" or what not! > Rather, my question is WHY do you see them as mutually > incompatible. Again where did I even hint these 2 are "mutually incompatible"? In fact, who know, some of the formulas about this unknown number might be logically equivalent to, say, CONT(PA)! > Is this some theorem of which I am unaware? > If so, can you provide a reference? Or is it some theorem > that you yourself have proved? If so, can you provide the > proof for our inspection? > > Is it only *syntactic* consistency proof that are so affected? Since I didn't say anything about them being incompatible, I obviously wouldn't able to answer these questions. > What about model-theoretic proofs of consistency. I seem > to recall that you do not accept them, but I could be wrong. > > Does this mutual incompatibility generalize, or is it something > specific to the natural numbers? > > >> [Imho, it could be said the the later epitomizes the impossibility of >> the former]. > > Can you explain how? I think I already did: they both connote impossibility of some syntactical proofs. So we just pick one as a representation (an epitome so to speak) of the other. > > >> That aside, it's actually my position that it's impossible to >> to syntactically prove a consistency: simply because the rules >> of inference won't let us do that; hence it's a _delusion_ that >> we could have any "sort of thing" that we could "accept as a >> proof of consistency"! > >> [That's why I'd would be surprised if in the past I had said something >> that has caused you to think there be a criteria to accept a proof >> of inconsistency]. > > As far as I am aware, you have not given the conditions under > which you would consider the consistency of a theory to be > proven. You're correct that I haven't, and I never will: because the conditions would be impossible to be met. That doesn't mean the following isn't a standard definition of syntactical consistency that you and I and everyone else would accept: consistency <-> not being syntactically inconsistent. > As far as proving INconsistency, I was not under the > impression that you disagreed with the usual method of > deriving a contradiction. That's correct that I've never disagreed: because the method of proving inconsistency is just a _finite FOL proof_, as I mentioned quite a few times here and elsewhere. > For the sake of completeness and > clarity, and because I suspect you meant "consistency" above > where you typed "inconsistency" would you clarify if/how you > consider it possible to prove a theory inconsistent. If you don't remember or have any doubt about my position, please allow me to clearly reiterate my position: It's impossible to have logically acceptable methods for proving a syntactical consistency. Hope that I've clarified my answer, position on your question about proving consistency. If that's what you meant to ask. (As for method of proving INconsistency, my position is that that's just like proving a FOL theorem, I've stated above).
From: Marshall on 5 Apr 2010 02:15 On Apr 4, 10:08 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Apr 4, 9:33 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Marshall wrote: > >>> On Apr 2, 5:32 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>> Let me put to rest the idea we know enough about the natural numbers, > >>>> to prove important thing such as the consistency of PA. I'll do that > >>>> by pointing out the existence of a specific unknown natural number. > >>> Why do you think the existence of a specific unknown number > >>> should have anything to do with consistency? > >> Because they (the syntactical proof of consistency and collectively > >> many formulas about this unknown natural) both connote the same thing > >> in meta level: impossibility of syntactical proof. If you can't prove > >> a certain formula related to this number, you can forget about proving > >> a consistency, syntactically speaking. > > > Both of these sentences are just a restatement of the position that > > I asked you for support of. My question is not, do you consider > > a proof of consistency of PA and the existence of an unknown natural > > number to be mutually incompatible. You have clearly stated so. > > Huh? I'm not quite following you here. First of, you clearly asked me > why the existence of a specific unknown natural number "should have > anything to do with consistency [proof]". And I gave you an answer for > that: their both connoting an "impossibility of [some] syntactical proof" > would be their "having anything to do" with each other. > > That's all I said here. And I didn't state anything about them being > "mutually incompatible" or what not! OK. So you mean that the two things have "something" to do with each other, but it's just a connotation, not any kind of logical result. > > Rather, my question is WHY do you see them as mutually > > incompatible. > > Again where did I even hint these 2 are "mutually incompatible"? When you said "If you can't prove a certain formula related to this number, you can forget about proving a consistency" I took that to mean that being unable to prove a certain formula is incompatible with a consistency proof. My mistake. Apparently what you are saying is that "you can forget about proving a consistency" but at the same time, proving consistency is still compatible. I guess I don't really understand that. > I think I already did: they both connote impossibility of some > syntactical proofs. So we just pick one as a representation > (an epitome so to speak) of the other. I have no idea what this is supposed to mean. > > As far as I am aware, you have not given the conditions under > > which you would consider the consistency of a theory to be > > proven. > > You're correct that I haven't, and I never will: because the conditions > would be impossible to be met. That doesn't mean the following isn't > a standard definition of syntactical consistency that you and I and > everyone else would accept: > > consistency <-> not being syntactically inconsistent. So you are saying that no theory can ever be proven to be consistent. (Also, how do you expect to convince me that the conditions for a consistency proof are impossible to meet without telling me what those conditions are?) Would you agree that this position is not one for which there is any mainstream acceptance? What about the theory with no axioms; is it impossible to prove that it is consistent? If it's not consistent, or not possible to prove it is consistent, what mechanism might admit an inconsistency? Surely it is not a logical conflict between some of its axioms? Are you then saying that FOL itself is not known to be consistent? What about propositional logic; is it possible to prove it consistent? What about boolean algebra? > > As far as proving INconsistency, I was not under the > > impression that you disagreed with the usual method of > > deriving a contradiction. > > That's correct that I've never disagreed: because the method of proving > inconsistency is just a _finite FOL proof_, as I mentioned quite a few > times here and elsewhere. And what if someone proved that a FOL proof of inconsistency for some particular theory is impossible? Or are you saying that it is impossible to prove that something is impossible to prove? If so, how do you know? Did you prove it? If you didn't, then how can you claim to know? If you did, I see a logical problem with your position. Marshall
From: Daryl McCullough on 5 Apr 2010 06:56 Newberry says... >On Apr 3, 9:51=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> It [the Godel statement for PA] can be proven. Just not in PA. > >Cool. So we know that the search for a proof of Goedel's sentence will >never terminate. Can we apply this knowledge to Diophantine equations? The Godel sentence is equivalent to the claim that a certain Diophantine equation has no solutions. So yes. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 5 Apr 2010 22:15 Marshall wrote: > On Apr 4, 10:08 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Apr 4, 9:33 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> Marshall wrote: >>>>> On Apr 2, 5:32 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>> Let me put to rest the idea we know enough about the natural numbers, >>>>>> to prove important thing such as the consistency of PA. I'll do that >>>>>> by pointing out the existence of a specific unknown natural number. >>>>> Why do you think the existence of a specific unknown number >>>>> should have anything to do with consistency? >>>> Because they (the syntactical proof of consistency and collectively >>>> many formulas about this unknown natural) both connote the same thing >>>> in meta level: impossibility of syntactical proof. If you can't prove >>>> a certain formula related to this number, you can forget about proving >>>> a consistency, syntactically speaking. >>> Both of these sentences are just a restatement of the position that >>> I asked you for support of. My question is not, do you consider >>> a proof of consistency of PA and the existence of an unknown natural >>> number to be mutually incompatible. You have clearly stated so. >> Huh? I'm not quite following you here. First of, you clearly asked me >> why the existence of a specific unknown natural number "should have >> anything to do with consistency [proof]". And I gave you an answer for >> that: their both connoting an "impossibility of [some] syntactical proof" >> would be their "having anything to do" with each other. >> >> That's all I said here. And I didn't state anything about them being >> "mutually incompatible" or what not! > > OK. So you mean that the two things have "something" to do > with each other, but it's just a connotation, not any kind of > logical result. But look at the connotation: "impossibility of [some] syntactical proof". Such an impossibility must have impact on logical reasoning and results, don't you think? >>> Rather, my question is WHY do you see them as mutually >>> incompatible. >> Again where did I even hint these 2 are "mutually incompatible"? > > When you said "If you can't prove a certain formula related to this > number, you can forget about proving a consistency" I took that to > mean that being unable to prove a certain formula is incompatible > with a consistency proof. My mistake. > > Apparently what you are saying is that "you can forget about > proving a consistency" but at the same time, proving consistency > is still compatible. I guess I don't really understand that. But where did I state anything like "proving consistency is still compatible"? In fact, "compatible" with what? I'm not sure I understand what the complaint here be. >> I think I already did: they both connote impossibility of some >> syntactical proofs. So we just pick one as a representation >> (an epitome so to speak) of the other. > > I have no idea what this is supposed to mean. It's akin to an equivalence class having 2 elements: you could choose one of them as a representation of the other. >>> As far as I am aware, you have not given the conditions under >>> which you would consider the consistency of a theory to be >>> proven. >> You're correct that I haven't, and I never will: because the conditions >> would be impossible to be met. That doesn't mean the following isn't >> a standard definition of syntactical consistency that you and I and >> everyone else would accept: >> >> consistency <-> not being syntactically inconsistent. > > So you are saying that no theory can ever be proven to be > consistent. That's correct. That's to say if you meant _syntactically_ consistent. > (Also, how do you expect to convince me that > the conditions for a consistency proof are impossible to > meet without telling me what those conditions are?) You yourself should be convinced: because the rules of inference doesn't have any provision to dis-prove a non-theorem, which would be needed to show the negation of syntactical inconsistency (i.e. consistency). > Would you agree that this position is not one for which > there is any mainstream acceptance? My agreement or disagreement is irrelevant here: we should go strictly by definitions of consistency, inconsistency, and of rules of inference. And the definition of rules of inference will not permit a proof of _syntactical_ consistency. > > What about the theory with no axioms; is it impossible to prove > that it is consistent? The assumed reasoning framework here is FOL=. So all theories have at least one axiom! In any rate, syntactically you can't use rules of inference to prove consistency, as explained above. > If it's not consistent, or not possible to > prove it is consistent, what mechanism might admit an > inconsistency? As having been explained, an inconsistency proof would be merely a finite syntactical proof. > Surely it is not a logical conflict between > some of its axioms? Are you then saying that FOL itself > is not known to be consistent? What about propositional > logic; is it possible to prove it consistent? What about > boolean algebra? Inconsistency and inconsistency here are meant to be of FOL formal systems. If we're not talking about formal systems then we're not talking about any other kind of "(in)consistency" in this context. (At least I'm not). So, for example, is your mentioned "boolean algebra" a formal system? >>> As far as proving INconsistency, I was not under the >>> impression that you disagreed with the usual method of >>> deriving a contradiction. >> That's correct that I've never disagreed: because the method of proving >> inconsistency is just a _finite FOL proof_, as I mentioned quite a few >> times here and elsewhere. > > And what if someone proved that a FOL proof of inconsistency > for some particular theory is impossible? No one can prove it's impossible to prove a formal system's inconsistency: if it's inconsistent, it's provable (in principle at least). > Or are you saying > that it is impossible to prove that something is > impossible to prove? I'm not sure if I had actually said that. The point I've said is we can't prove syntactical consistency of a formal system via rules of inference. > If so, how do you know? Did you prove it? > If you didn't, then how can you claim to know? If you did, I see > a logical problem with your position. > > > Marshall
From: Nam Nguyen on 10 Apr 2010 00:46
Nam Nguyen wrote: > Jim Burns wrote: > >> I think it would be very useful to me in understanding >> what you are trying to accomplish if you were >> to give a summary of the best arguments AGAINST your >> positions. > > As promised I'll summarize what I think as the best arguments > against my positions. > *** > > Imho, the 3 major and best arguments against my belief, that the > nature of FOL reasoning is that of relativity or of being subjective, > are the following objections: > > (O1) The Universality Objection: > > In this objection, the correctness in reasoning under one logical > framework should be _universally constant_ and shouldn't be a > function of individual subjective beliefs or knowledge. My claiming > on the relativity nature of FOL reasoning seems to violate this > natural and unobjectionable, say, "sanctity". > > (O2) The Philosophy Objection: > > In this objection, the ideas such that there are formulas written > in the language of arithmetic that can be neither arithmetically > true nor false are just philosophical ideas and thus can't be a > basis to attack the current FOL reasoning. > > (O3) The Ordinary Mathematics Objection: > > This objection seems to be a cross-breed between O1 and O2. In this > objection, FOL reasoning is build upon the ordinary mathematical > knowledge that *in principle* should be universally _self evident_ > to all who are trained or study mathematics. As such FOL reasoning > should be universally the same and should _not_ be subjective or > relativistic. > > Again, these are only my thoughts of what the arguments against my position > be. It would certainly be helpful if those who oppose my position could > further clarify in technical clarity what they perceive are problems in my > positions. I also don't mind in subsequent posts to further defend my > position or to provide more counter-arguments. I'm still hoping to hear some responses, from Jim, my opponents, or from anyone. |