How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
in [Logic]
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From: Aatu Koskensilta on 25 Jun 2010 05:14 Charlie-Boo <shymathguy(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. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Alan Smaill on 25 Jun 2010 05:19 Aatu Koskensilta <aatu.koskensilta(a)uta.fi> writes: > Charlie-Boo <shymathguy(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. It is of course more educational to work this out for oneself. -- Alan Smaill
From: Jesse F. Hughes on 25 Jun 2010 10:21 Charlie-Boo <shymathguy(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 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). 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. -- 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
From: Jesse F. Hughes on 27 Jun 2010 09:07 Transfer Principle <lwalke3(a)lausd.net> writes: > Hughes will undoubtedly disagree with me, but I find the > arrival of all these opponents of ZFC at the same time > simply hilarious... I don't know about hilarious, but it is entertaining. I imagine that you find it hilarious because in your addled brain, the fact that so many "opponents of ZFC" are on the group at once indicates the old guard is embattled, that the oppressors have their backs against the wall. I find it entertaining because some of the things these folks say are funny and that it's sometimes interesting to see how they defend their inconsistencies, but it's not really intellectually stimulating. It's a cheap feeling of victory to get Tony to admit (1) Every element of N+ is a finite number. (2) Tav is an element of N+. (3) Tav is not a finite number. But, Walker, you really have the wrong impression of me. I come to sci.math mostly to read the cranks. I'm not proud of that fact, but it's true. They are not an enemy threatening to topple my deeply held beliefs. They're just eccentric, entertaining folks -- even if I sometimes become annoyed at them, I surely don't wish they would go away[1]. Footnotes: [1] To be sure, I don't enjoy all cranks. Matter of taste, really. -- "Now I realize that he got away with all of that because sci.math is not important, and the rest of the world doesn't pay attention. Like, no one is worried about football players reading sci.math postings!" -- James S. Harris on jock reading habits
From: Jesse F. Hughes on 27 Jun 2010 09:12
Charlie-Boo <shymathguy(a)gmail.com> writes: > On Jun 26, 10:41 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Charlie-Boo <shymath...(a)gmail.com> writes: >> > On Jun 25, 10:21 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> >> 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! >> >> Yes. The usual language of ZFC does not have a successor function >> symbol, while the language of PA does. Thus, we must extend *the >> language* of ZFC and also add a defining axiom for the successor >> function. > > "add an axiom" > > Yikes! Yikes! You might want to learn about conservative extensions of a theory. Any time you add a function symbol to a language, you must also add a defining axiom to the theory if you want the function to be defined. In "good" cases, one can prove that the extension is conservative. Utterly standard. Wikipedia has pages on both "Conservative extensions" and "Extensions by definitions". -- "Witty adolescent banter relies highly on the use of 'whatever.' Anyone out of high school forced to watch more than an hour of 'Laguna Beach' might possibly feel the urge to beat themselves about the head with a large stick." -- NY Times on an MTV reality show |