How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: MoeBlee on 27 Jun 2010 15:44 On Jun 26, 6:38 am, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote: > The theory ZF-Inf+~Inf clearly proves ~Inf ("Infinite sets do not > exist"). ~Inf is not (1) "there does not exist an infinite set", but rather ~Inf is (2) "there does not exist a successor inductive set". I don't know that there is a proof in Z set theory of the equivalence of (1) and (2). However, in ZF-Inf we can prove that equivalence, but it does take a bit of argument. > This proof obviously implies that "There does not exist a > model for PA", You get credit for skillful legerdemain here, but nothing more. Let's call ZF-Inf+~Inf by the name 'Y'. Now Y proves that there does not exist ANY theory (in the ordinary sense in which first order PA is a theory). Moreover, Y proves that there does not even exist any first order LANGUAGE to be the language of a theory. This is all obvious (once you think about it for a moment and are not distracted too much by your logical legerdemain): A a first order language has an infinite set of variables, and a theory is a certain kind of infinite set of sentences, so if there are no infinite sets, then there are no first order languages and no theories (in the sense in which first order PA is a theory). Your meta-theory Y proves that there IS NO object that satisfies the description we provide the rubric 'PA'. Any argument you base on theory Y proving anything about PA (with 'PA' defined in some ordinary way) is just an exercise in vacuous reasoning. Not only does Y prove that anything that fulfills the description we give to PA does not have a model, but Y also proves that anything that fulfills the description we give to PA DOES have a model, since Y proves that THERE IS NO OBJECT that fulfills the description we give to PA. Of course, you could eschew vacuous reasoning as I just gave, but then you're NOT using Y as the meta-theory, since Y=ZF-Inf+~Inf deploys classical first order logic. If you wish to argue from some OTHER logic for the meta-theory then that meta-theory is not ZF-Inf+~Inf. I expect that if you answer this point, you will do so with even more of your confusions about the basics of mathematical logic. In that case, I likely will not bother to serve as your nurse to clean up your mess. MoeBlee
From: MoeBlee on 27 Jun 2010 15:48 On Jun 26, 5:56 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > On Jun 24, 4:19 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 24, 12:47 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > > On Jun 14, 11:45 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > > > Charlie-Boo <shymath...(a)gmail.com> writes: > > > > > Since PA can't prove something as simple as that, how could anyone be > > > > > so stupid as to claim ZFC/PA is a good basis for all of our ordinary > > > > > math? > > > > > Who makes this claim? > > > > MoeBlee > > > What I've said on the subject is in my own words and is not properly > > simplified to "ZFC/PA is a good basis for all of our ordinary math", > > especially as I don't know what is supposed to be indicated by 'ZFC/ > > PA' in such slash notation. (PA is embeded in ZFC, of course.) > > > For example, recently I said, "The common claim is that ZFC > > axiomatizes all (or virtually all) ordinary mathematics. " > > > But I did not say that I personally make that common claim. I > merely > > said what the common claim IS; I didn't say that it is also a > claim > > that I make. > > So you don't claim it but you do claim that many people claim it. Right. I regard the claim as eminently plausible, from what I've seen; but there is a vast amount of ordinary mathematics that I have not checked for this claim. > Do > you claim that many people claim that they claim it? Or many that > claim that they don't claim it (e.g. you)? And do you claim that > there are many people who claim that you claim that they claim that > you claim it? Come on, if you're going to engage my responses and my time, please don't waste my time with unfunny silliness. MoeBlee
From: MoeBlee on 27 Jun 2010 15:50 On Jun 26, 6:04 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > On Jun 24, 5:38 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 24, 3:58 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > > On Jun 24, 4:19 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > > On Jun 24, 12:47 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > > > > On Jun 14, 11:45 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > > > > > Charlie-Boo <shymath...(a)gmail.com> writes: > > > > > > > Since PA can't prove something as simple as that, how could anyone be > > > > > > > so stupid as to claim ZFC/PA is a good basis for all of our ordinary > > > > > > > math? > > > > > > > Who makes this claim? > > > > > > MoeBlee > > > > > What I've said on the subject is in my own words and is not properly > > > > simplified to "ZFC/PA is a good basis for all of our ordinary math", > > > > especially as I don't know what is supposed to be indicated by 'ZFC/ > > > > PA' in such slash notation. (PA is embeded in ZFC, of course.) > > > > > For example, recently I said, "The common claim is that ZFC > > > > axiomatizes all (or virtually all) ordinary mathematics. " > > > > > But I did not say that I personally make that common claim. I > > > merely > > > > said what the common claim IS; > > > > And that being a common claim is what I claimed, so you claimed the > > > same thing that I claimed. > > > You claim that they are the same claim, though I noted specific > > differences. I don't need to argue whether they are the same, but > only > > I note that I stand by my own wording > > So you would agree that "I always tell the truth."? (Ambiguity...) > > So you might say "I always tell the truth. As far as what I say, you can just look at my posts. I don't see the point of your silliness above. MoeBlee
From: MoeBlee on 27 Jun 2010 15:58 On Jun 26, 11:24 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > This has come up time and time again. I myself have claimed that > the theory ZF-Infinity+~Infinity proves that every set is finite, > and someone (usually MoeBlee or Rupert) points out that this > theory only proves that there's no _successor-inductive_ set > containing 0, not that there is no infinite set. Not as I recall. Rather Z-Infinity+~Infinity seems not to prove "there does not exist an infinite set". But ZF-Infinity+~Infinity DOES prove "there does not exist an infinite set". (Infinite defined as "not equinumerous with a natural number".) > And every time this comes up, I want to say _fine_ -- so if > ZF-Inf+~Inf _doesn't_ prove that every set is finite, But, as I just mentioned, I don't know who says that is the case. MoeBlee
From: R. Srinivasan on 27 Jun 2010 16:25
On Jun 28, 12:44 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 26, 6:38 am, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote: > > > The theory ZF-Inf+~Inf clearly proves ~Inf ("Infinite sets do not > > exist"). > > ~Inf is not (1) "there does not exist an infinite set", but rather > ~Inf is (2) "there does not exist a successor inductive set". I don't > know that there is a proof in Z set theory of the equivalence of (1) > and (2). However, in ZF-Inf we can prove that equivalence, but it does > take a bit of argument. > > > This proof obviously implies that "There does not exist a > > model for PA", > > You get credit for skillful legerdemain here, but nothing more. > > Let's call ZF-Inf+~Inf by the name 'Y'. Now Y proves that there does > not exist ANY theory (in the ordinary sense in which first order PA is > a theory). Moreover, Y proves that there does not even exist any first > order LANGUAGE to be the language of a theory. This is all obvious > (once you think about it for a moment and are not distracted too much > by your logical legerdemain): A a first order language has an infinite > set of variables, and a theory is a certain kind of infinite set of > sentences, so if there are no infinite sets, then there are no first > order languages and no theories (in the sense in which first order PA > is a theory). Your meta-theory Y proves that there IS NO object that > satisfies the description we provide the rubric 'PA'. > > Any argument you base on theory Y proving anything about PA (with 'PA' > defined in some ordinary way) is just an exercise in vacuous > reasoning. Not only does Y prove that anything that fulfills the > description we give to PA does not have a model, but Y also proves > that anything that fulfills the description we give to PA DOES have a > model, since Y proves that THERE IS NO OBJECT that fulfills the > description we give to PA. > > Of course, you could eschew vacuous reasoning as I just gave, but then > you're NOT using Y as the meta-theory, since Y=ZF-Inf+~Inf deploys > classical first order logic. If you wish to argue from some OTHER > logic for the meta-theory then that meta-theory is not ZF-Inf+~Inf. > > I expect that if you answer this point, you will do so with even more > of your confusions about the basics of mathematical logic. In that > case, I likely will not bother to serve as your nurse to clean up your > mess. > > I was wondering why it took so long for you to make an appearance here. My arguments are based entirely on FOL and I am not bringing in any theory from NAFL as the metatheory of (classical) PA. So let us apply your example of "vacuous reasoning". The theory PA is about numbers. It certainly does not know what "PA" is. Anything that you claim about PA proving the consistency of PA is just utter hogwash, since PA proves that "PA" does not exist, So Godel What I outlined is simple straightforward stuff. By "metatheory" of PA, I mean a theory in which PA is interpretable and therefore automatically encodes sentences like con(PA). So certainly, when you use suitable coding, the theory ZF-Inf+~Inf does know what PA is. A second requirement is that the metatheory should suitably extend the language of PA (under suitable "coding", of course) to either allow for objects that can be interpreted as models of PA or deny the existence of such objects. The theory ZF-Inf+~Inf takes the latter route. It is all about *coding*. I am surprised that you failed to see this. Or are you saying that "coding" was Godel's "logical legerdemain" (whatever that means)? The proof in the theory ZF-Inf +~Inf that infinite sets do not exist encodes (among other things) that models of PA cannot exist. That is all I need to calll ZF-Inf +~Inf as a metatheory of PA. It is not necessary that a metatheory of PA should treat PA as an explicit mathematical object in its language. If you think so, that is your confusion. RS |