How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: Charlie-Boo on 26 Jun 2010 20:56 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. 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? See my "1,000 Paradoxes" post where I list 1,000 English sentences that show what a real formalization of the Liar contains. "This is true of 'It is not true of this.'" etc. C-B > And I didn't say anything about ZFC being a "good" basis. Good in what > sense? ZFC has certain merits and (arguably) certain drawbacks. It may > be a suitable theory in certain ways, but I did not claim that it is > simply "good" as a basis. > > Also, I allowed that a reasonable view of the common claim may include > that only VIRTUALLY all of ordinary mathematics may be axiomatized by > ZFC. > > MoeBlee
From: Charlie-Boo on 26 Jun 2010 21:04 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."? C-B "I made a misstatement and I stand by all my misstatements." - Dan Quayle and I don't obligate myself to > defend your wording. > > MoeBlee- Hide quoted text - > > - Show quoted text -
From: Charlie-Boo on 26 Jun 2010 21:09 On Jun 24, 6:04 pm, George Greene <gree...(a)email.unc.edu> wrote: > On Jun 14, 11:42 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > ZFC/PA is supposed to do all ordinary mathematics*. But it is easy to > > prove that PA is consistent (its axioms and rules preserve truth) yet > > (by Godel-2) PA can't do such a simple proof as that. > > So what? ZFC can prove it. > ZFC can't prove that ZFC is consistent, but it CAN AND DOES prove > that PA is consistent. This is why you can't say that "ZFC/PA > doesn't prove PA is consistent." "ZFC/PA" is just a meaningless > locution in any case. Since ZFC/PA is meaningles, how does ZFC show that we can't say that "ZFC/PA doesn't prove PA is consistent."? How can a system of logic show that an expression that contains meaningless syntax can't be said (is not truthful)? > ZFC is one thing. PA is another. And CBL is still another. However, CBL proves theorems with proofs that are about 1% the size of those published, while ZFC and PA take about 10 times the size published. So which is best? C-B
From: Charlie-Boo on 26 Jun 2010 21:28 On Jun 25, 5:19 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > Aatu Koskensilta <aatu.koskensi...(a)uta.fi> writes: > > Charlie-Boo <shymath...(a)gmail.com> writes: > > >> Who has proved PA consistent using ZFC? If it were possible then I > >> assume someone would have done it. It certainly would be a very > >> educational exercise. > > > So why not have a try at it? You'll find all the details you need in any > > decent text. > > Not to mention that it has been outlined several times in sci.logic. Reference to a post with it? What of ZFC's set theoretic axioms is necessary - especially not bookkeeping ones like sets existing that are used throughout PA and are not needed in every proof? That is, I see little added by ZFC's axioms over PA's which are stolen anyway in the form of "definitions" that N has certain properties i.e. satisfies the PA axioms. So the question is what does ZFC provide that is needed that PA does not (implicitly in that it does mathematics at worst)? > It is of course more educational to work this out for oneself. Well, the first question is which proof to use. Then there is the question of how to formalize it. So it's at least 2 distinct steps. C-B > -- > Alan Smaill
From: Charlie-Boo on 26 Jun 2010 21:34
On Jun 25, 10:21 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > On Jun 24, 6:04 pm, George Greene <gree...(a)email.unc.edu> wrote: > >> On Jun 14, 11:42 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > > >> > ZFC/PA is supposed to do all ordinary mathematics*. But it is easy to > >> > prove that PA is consistent (its axioms and rules preserve truth) yet > >> > (by Godel-2) PA can't do such a simple proof as that. > > >> So what? ZFC can prove it. > >> ZFC can't prove that ZFC is consistent, but it CAN AND DOES prove > >> that PA is consistent. This is why you can't say that "ZFC/PA > >> doesn't prove PA is consistent." "ZFC/PA" is just a meaningless > >> locution in any case. > >> ZFC is one thing. PA is another. > > > PA is a subset of ZFC, so I emphasize that by calling it ZFC/PA (it > > makes more sense to distinguish the two anyway.) This is besides the > > point. Who has proved PA consistent using ZFC? If it were possible > > then I assume someone would have done it. It certainly would be a > > very educational exercise. > > > In any case, it shows the weakness of PA. I added ZFC as that is so > > popular. > > No, I think you have a good point Thanks. > and an interesting new form of > argument. I'm gonna try it myself. > > People say that atoms are made up of subatomic particles. But you > can't make atoms up from protons, because they repeal each other. So > why would people think this? > > This is a great argument, because the class of protons is a subset of > the class of subatomic particles, just as the theorems of PA are a > subset of the theorems of ZFC (with suitable extension of the language > of ZFC). "with suitable extension of ZFC" Yikes! > I are as smart as Charlie. > > Final hint, Charlie: if someone says that ZFC suffices, and you show > that a subset of ZFC does *not* suffice, then you haven't refuted > their claim. my goodness C-B > -- > Jesse F. Hughes > "[M]oving towards development meetings for new release class viewer 5.0 > and since [I]'m the only developer, easy to schedule." > --James S. Harris tweets on code development- Hide quoted text - > > - Show quoted text - |