How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent(EASY PROOF)
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From: Nam Nguyen on 1 Jul 2010 14:40 MoeBlee wrote: > On Jul 1, 9:01 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> MoeBlee wrote: >>> On Jun 29, 10:47 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> Aatu Koskensilta wrote: >>>>> Frederick Williams <frederick.willia...(a)tesco.net> writes: >>>>>> Yes, you can: take Gentzen's proof (or Ackermann's etc) and formalize >>>>>> it in ZFC. >>>>> This is a pretty silly way of proving the consistency of PA in set >>>>> theory. >>>>> That PA is consistent is a triviality. >>>> In what formal system is this triviality in? >>> It's a theory of Z-R, for example. Whether it's "trivial" to prove in >>> Z-R depends on what strikes one as trivial. >>>> (Iow, you didn't mean >>>> it's a fact that PA is syntactically consistent, did you?) >>> Consistent IS syntactically consistent. >> But there's also such thing as relative consistency proof! > > Yes, of course. I don't know why you're excited about that fact > though. No emotion on my part. It's just in technical arguments you should use the terminologies precisely. PA's (in)consistency, by definition of (in)consistency, is a _fact_. If you don't know that fact then precisely state so. Don't just use a relative consistency proof and "somehow" bootstrap it into a "fact"! > >> For example, >> from T = {Ax[xex] /\ ~Ax[xex]}, it's a triviality to prove the consistency >> of PA, > > Sure, as long as there is some sentence in the language of T that we > read as "PA is not consistent". Of course, such a proof does not in > itself give evidence that there is a PA proof of a formula P&~P. > Rather, such a proof gives evidence merely that in T there is a > certain derivation of a formula that we are reading as "PA is not > consistent". You're missing the point though: _you_ don't know how to prove the _fact_ of (in)consistency of PA; and in general relative consistency proofs aren't interesting because it will NOT confirm that _fact_! > >> but should I proclaim that PA is consistent, as in, "that PA is >> consistent is a triviality", as Aatu put it? > > Right, we agree you should not take such a proof as evidentiary in > that way. But, just to be clear (since I'm not sure exactly what > you're saying) Aatu is not claiming that you should. > >> The question I had for him was a clarification request to see if he meant >> PA is really consistent, > > Yes, he means that PA is consistent, really consistent. From his recent responses, I think he had referred to a relative consistency proof. > >> or if he meant that was just a relative consistency >> proof he had referred to. > > The above you referred to is not a relative consistency. > > A relative consistency is of the form: > > T |- G consistent -> G* consistent > > The proof you mentioned is of the form: > > T |- G consistent. > > Anyway, Aatu is not saying just that there exists a relative > consistency proof nor just that, say, ZF or some other formal system > proves Con(PA), but rather he's saying that PA IS consistent. He's > saying that aside from whatever FORMAL proofs, PA is consistent - > PERIOD. A crank would "say" anything too! But I've never believed Aatu is a crank so where's his _proof_, in FOL level or meta level? Oh, but you're going to explain the "proof" right below, I see. > His basis is for that is not a FORMAL proof, but rather his > conviction that the axioms of PA are true (and not even in confined to > a FORMAL model theoretic sense of truth, but rather that the axioms > are simply true about the natural numbers, as we (editorial 'we') > understand the natural numbers even aside from any formalization. Let me see: his proof - isn't based on rules of inference and axioms - isn't based on "model theoretic sense of truth" - is merely based on _conviction_ that "the axioms of PA are true" and our intuitive knowledge of the natural numbers "aside from any formalization". Wow! A lot of people have "convictions" and "intuitions" in reasoning too you know! (Including some well known cranks in the 2 fora!) Seriously, if you and he don't abide to the strictness of FOL proof and FOL language model definition, then (at minimum) you should have NOT asserted PA's consistency since that's a simple straightforward notion in FOL that you either don't know how to prove it, or prove it in the guidelines and definition of FOL (in)consistency. Conviction and intuitions _might_ help reasoning but is in no way a replacement of reasoning! > > Haven't you read Franzen's incompleteness book? I read part of the book. So? Would wrong become right somehow, or vice versa? > >> (You should read people's conversation more carefully, before jumping to >> conclusion whether or not people understand this or that.) > > I didn't post anything that shows lack of context of the conversation. Why did you post an _incorrect_ definition of formal system consistency?
From: Nam Nguyen on 1 Jul 2010 15:10 Nam Nguyen wrote: > MoeBlee wrote: >> On Jul 1, 9:01 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>> MoeBlee wrote: >>>> On Jun 29, 10:47 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>> Aatu Koskensilta wrote: >>>>>> Frederick Williams <frederick.willia...(a)tesco.net> writes: >>>>>>> Yes, you can: take Gentzen's proof (or Ackermann's etc) and >>>>>>> formalize >>>>>>> it in ZFC. >>>>>> This is a pretty silly way of proving the consistency of PA in set >>>>>> theory. >>>>>> That PA is consistent is a triviality. >>>>> In what formal system is this triviality in? >>>> It's a theory of Z-R, for example. Whether it's "trivial" to prove in >>>> Z-R depends on what strikes one as trivial. >>>>> (Iow, you didn't mean >>>>> it's a fact that PA is syntactically consistent, did you?) >>>> Consistent IS syntactically consistent. >>> But there's also such thing as relative consistency proof! >> >> Yes, of course. I don't know why you're excited about that fact >> though. > > No emotion on my part. It's just in technical arguments you should > use the terminologies precisely. PA's (in)consistency, by definition > of (in)consistency, is a _fact_. If you don't know that fact then precisely > state so. Don't just use a relative consistency proof and "somehow" > bootstrap it into a "fact"! > >> >>> For example, >>> from T = {Ax[xex] /\ ~Ax[xex]}, it's a triviality to prove the >>> consistency >>> of PA, >> >> Sure, as long as there is some sentence in the language of T that we >> read as "PA is not consistent". Of course, such a proof does not in >> itself give evidence that there is a PA proof of a formula P&~P. >> Rather, such a proof gives evidence merely that in T there is a >> certain derivation of a formula that we are reading as "PA is not >> consistent". > > You're missing the point though: _you_ don't know how to prove the _fact_ > of (in)consistency of PA; and in general relative consistency proofs > aren't interesting because it will NOT confirm that _fact_! > >> >>> but should I proclaim that PA is consistent, as in, "that PA is >>> consistent is a triviality", as Aatu put it? >> >> Right, we agree you should not take such a proof as evidentiary in >> that way. But, just to be clear (since I'm not sure exactly what >> you're saying) Aatu is not claiming that you should. >> >>> The question I had for him was a clarification request to see if he >>> meant >>> PA is really consistent, >> >> Yes, he means that PA is consistent, really consistent. > > From his recent responses, I think he had referred to a relative > consistency proof. > >> >>> or if he meant that was just a relative consistency >>> proof he had referred to. >> >> The above you referred to is not a relative consistency. >> >> A relative consistency is of the form: >> >> T |- G consistent -> G* consistent >> >> The proof you mentioned is of the form: >> >> T |- G consistent. >> >> Anyway, Aatu is not saying just that there exists a relative >> consistency proof nor just that, say, ZF or some other formal system >> proves Con(PA), but rather he's saying that PA IS consistent. He's >> saying that aside from whatever FORMAL proofs, PA is consistent - >> PERIOD. > > A crank would "say" anything too! But I've never believed Aatu is a crank > so where's his _proof_, in FOL level or meta level? Oh, but you're going > to explain the "proof" right below, I see. > >> His basis is for that is not a FORMAL proof, but rather his >> conviction that the axioms of PA are true (and not even in confined to >> a FORMAL model theoretic sense of truth, but rather that the axioms >> are simply true about the natural numbers, as we (editorial 'we') >> understand the natural numbers even aside from any formalization. > > Let me see: his proof > > - isn't based on rules of inference and axioms > - isn't based on "model theoretic sense of truth" > - is merely based on _conviction_ that "the axioms of PA are > true" and our intuitive knowledge of the natural numbers > "aside from any formalization". > > Wow! A lot of people have "convictions" and "intuitions" in reasoning too > you know! (Including some well known cranks in the 2 fora!) > > Seriously, if you and he don't abide to the strictness of FOL proof and > FOL language model definition, then (at minimum) you should have NOT > asserted PA's consistency since that's a simple straightforward > notion in FOL that you either don't know how to prove it, or prove > it in the guidelines and definition of FOL (in)consistency. > > Conviction and intuitions _might_ help reasoning but is in no way > a replacement of reasoning! > >> >> Haven't you read Franzen's incompleteness book? > > I read part of the book. So? Would wrong become right somehow, or vice > versa? > >> >>> (You should read people's conversation more carefully, before jumping to >>> conclusion whether or not people understand this or that.) >> >> I didn't post anything that shows lack of context of the conversation. > > Why did you post an _incorrect_ definition of formal system consistency? It's not like I like to bash the concept of the natural numbers and the intuitive knowledge of PA for no reason. Consider the 2 below formal systems PA' and PA'': PA' = PA + (1) PA'' = PA + ~(1) If PA is consistent, what can you say about the (in)consistency of the 2 above? And if your intuition of the natural numbers is precise, what's your conviction about your intuition about a model of any of the 2 systems above? You and he may as well concede now, and save yourself some effort trying to answer these questions.
From: Nam Nguyen on 1 Jul 2010 16:48 MoeBlee wrote: > On Jul 1, 1:40 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> No emotion on my part. It's just in technical arguments you should >> use the terminologies precisely. > > As precise as is required to communicate. Absolute precision in > English is elusive. Yeah right. You said: > I didn't post ANY definition of the consistency of a formal system. I > posted a definition of consistency of a set of formulas. So, according to _your_ "precise" definition, S = {GC, cGC} is precisely a consistent set of formulas, right? No wonder it's impossible to have fruitful conversation with you on technical foundational matters.
From: herbzet on 1 Jul 2010 16:50 Nam Nguyen wrote: > You're missing the point though: _you_ don't know how to prove the _fact_ > of (in)consistency of PA; and in general relative consistency proofs > aren't interesting because it will NOT confirm that _fact_! Just the facts, ma'am. - Sgt. Joe Friday, "Dragnet" - THAT'S THE FACK, JACK! - Pvt. John Winger, "Stripes" -
From: Nam Nguyen on 3 Jul 2010 00:10
Transfer Principle wrote: > On Jul 1, 9:25 am, MoeBlee <jazzm...(a)hotmail.com> wrote: >> On Jul 1, 9:01 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>> or if he meant that was just a relative consistency >>> proof he had referred to. >> Anyway, Aatu is not saying just that there exists a relative >> consistency proof nor just that, say, ZF or some other formal system >> proves Con(PA), but rather he's saying that PA IS consistent. He's >> saying that aside from whatever FORMAL proofs, PA is consistent - >> PERIOD. His basis is for that is not a FORMAL proof, but rather his >> conviction that the axioms of PA are true (and not even in confined to >> a FORMAL model theoretic sense of truth, but rather that the axioms >> are simply true about the natural numbers, as we (editorial 'we') >> understand the natural numbers even aside from any formalization. > > But this raises an interesting point here. > > If Aatu can say that PA is consistent, _period_, without any formal > proof whatsoever, then why can't Nguyen believe that PA is > inconsistent, > _period_, without formal proof? For that matter, why can't Herc > believe > that there exist only countably many reals, _period_, without formal > proof, or Srinivasan believe that Infinity is false, _period_, without > formal > proof, or WM believe that certain large naturals don't exist, > _period_, > without formal proof? I agree with you in the above: there's a degree of being double standard that Moeblee and other "standard theorists" have exercised: they'd blast people as talking nonsense if people don't conform to standard logics in arguing while when it's their turn to conform to definitions of consistency and language models in proving, say, PA's syntactical consistency, it'd be perfectly OK for them to _ignore formal proofs and just use mere intuitions_! The most interesting question is why MoeBlee and those "standard theorists" never admit they're just being philosophical about PA's consistency, while they blatantly admit that they've gone astray from rigorousness of reasoning? I mean MoeBlee said above: "aside from whatever FORMAL proofs, PA is consistent". Then why did the founders of FOL invent FORMAL proofs in the first place? So that MoeBlee and others in that group could throw these formal proofs away like boring toys, whenever they feel they'd like to? If that's isn't being double standard in arguing, reasoning then I don't know what is. |