How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: Frederick Williams on 1 Jul 2010 15:50 Charlie-Boo wrote: > No, PA and ZFC both have Peano's Axioms. What formulations of first order PA and ZFC have _any_ of their non-logical axioms in common? None. Why? Well, for starters, all of the proper axioms of ZFC have the predicate symbol $\in$ in them, none of PA's axioms do because $\in$ is not even in the language of PA > The only difference is the > universal set, which is N in PA There are no sets, universal or otherwise, in first order PA. > and sets in ZFC. That's why it is > expressed differently - different alphabets as well. -- I can't go on, I'll go on.
From: Frederick Williams on 1 Jul 2010 16:02 Charlie-Boo wrote: > The axiom of infinity was added to give ZFC the capabilities of PA. No, the axiom of infinity is there so that ZFC may talk about infinite sets: omega and a whole load of others. Set theory begins with Cantor and has some famous difficulties that arise in connection with infinite sets. Zermelo's purpose was to axiomatize away the difficulties but keep the infinite sets. > Now, "to give" here refers to the state of mind of its author, so > let's not be psychotic. Let me say that it seems that's all it gives > you - that's all it's used for. So mathematicians have nothing to say about uncountable sets? > But all of you ZFC-ites can say if it > has been used for anything else. -- I can't go on, I'll go on.
From: MoeBlee on 1 Jul 2010 16:04 On Jul 1, 1:40 pm, Nam Nguyen <namducngu...(a)shaw.ca> 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. As precise as is required to communicate. Absolute precision in English is elusive. > PA's (in)consistency, by definition > of (in)consistency, is a _fact_. I guess you mean that if PA is consistent then that is a fact and that if PA is inconsistent then that is a fact. (?) > If you don't know that fact then precisely > state so. I haven't opined on what I know or don't know to be a fact in that matter. > Don't just use a relative consistency proof and "somehow" > bootstrap it into a "fact"! Again, you're excited but I can't fathom why. I never used anything about a relative consistency proof to say anything about facts in the matter of the consistency of PA. > >> 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_! I find it impossible to communicate with you. What I said above is, as far as I can tell, pretty much the same thing as you are saying yourself in your response, so I'm not "missing" the point at all. I'm sorry, Nam, after years of posting with you, and also reading your posts with others, I just have to conclude that communication on these subjects can't be achieved with you. > >> 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. READ what Aatu has written about this about a hundred times. He ESCHEWS appealing to a relative consistency proof as a fundamental basis for believing that PA is consistent. > >> 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. WHAT proof? Proof of WHAT? Proof of the consistency of PA? There are many. But Aatu is saying such proofs are NOT the basis for his conviction that PA is consistent. > > 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". I didn't say it is a PROOF. Why are you not LISTENING? > Wow! A lot of people have "convictions" and "intuitions" in reasoning too > you know! (Including some well known cranks in the 2 fora!) If you do not find Aatu's postings on this subject to be convincing, along with the writings of such people as Franzen, then fine. > Seriously, if you and he don't abide to the strictness of FOL proof and > FOL language model definition, Good thing you said "IF", because I didn't say that Aatu's viewpoint is my own. I'm just stressing what his viewpoint IS. > > Haven't you read Franzen's incompleteness book? > > I read part of the book. So? Would wrong become right somehow, or vice > versa? What would happen is that you would get a better understanding of his point of view. > >> (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? I didn't post ANY definition of the consistency of a formal system. I posted a definition of consistency of a set of formulas. MoeBlee
From: MoeBlee on 1 Jul 2010 16:09 On Jul 1, 2:30 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > The point is that ZFC would have to have an axiom other than Infinity > (i.e. PA) that is necessary to prove PA consistent in order for it to > be impossible in PA and possible in ZFC. But again you don't show > that. Talking to myself: PLEASE MoeBlee don't waste a second more of your time trying to get through to the ineducable Charlie-Boo. Listen, MoeBlee, no matter how much you explain, no matter how detailed or general, Charlie-Boo will still persist with yet more of his confusions over even the most simple matters such as the difference between Z set theory and first order PA. MoeBee
From: MoeBlee on 1 Jul 2010 16:18
On Jul 1, 2:45 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > It's not a theorem until proven. It's not KNOWN to be a theorem until a proof has been witnessed. A claim of theoremhood is not justified until proof has been witnessed. Anyway, I posted in the very post you're QUOTING an (informal, of course) set theoretic proof that if a theory has a model then that theory is consistent. Damn! Here it is as you QUOTED yourself: > > A theory is a set of sentences (all in a language) closed under > > entailment. If a theory T is inconsistent, then there is a sentence P > > such that both P and ~P are in T. But P is true in model M iff ~P is > > false in model M, and no sentence is both true and false in a given > > model M (by the definition-by-recursion function that maps sentences > > to true or (exclusive or) to false per a model). So if a theory is > > inconsistent, then the theory has no model (lest there be a sentence P > > that is both true and false in the model, which is impossible). There it is. It takes but two minutes, just thinking about the definitions, to come up with the above. You REALLY couldn't or wouldn't even try? > > > Didn't you read my response? Hinman doesn't refer to ZFC's axioms at > > > all in his proof. > > > The axioms are used in the various steps leading up to the proof. > > That's how mathematics works. A proof of a theorem may rely on > > previously proven theorems. > > > > He even admits that. > > > What specific quote do you have in mind? Did you miss seeing that question? Would you please say exactly what statement by Hinman you construe as "admitting" that he doesn't "refer" (or use) axioms of ZF? (The C in ZFC is not needed for this particular task; for that matter, the F is not needed.) MoeBlee |