How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
in [Math]
|
Prev: ? theoretically solved
Next: How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
From: Charlie-Boo on 26 Jun 2010 23:04 On Jun 14, 10:41 pm, George Greene <gree...(a)email.unc.edu> wrote: > On Jun 14, 11:42 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > ZFC/PA > > You're AN IDIOT. Atta said you're an idiot. He said: "This doesn't make much sense." That means you're an idiot, not me. Who's smarter, you or Atta? (His name is a zillion 9,999,999,999 matches on Google!!) C-B Ref.s: "This doesn't make much sense." - Atta > ZFC IS ONE thing. > PA IS ANOTHER, SIMPLER thing. > ZFC is A SET theory. > PA is a theory OF ARITHMETIC. > > First-order theories in general cannot prove THEMSELVES consistent, > so first-order PA cannot prove that first-order PA is consistent. > First-order ZFC, however, IS A STRONGER THEORY (it has an > axiom of infinity), SO IT CAN AND DOES prove that PA is consistent. > > Your problem is that you presumed to talk about ZFC/PA like it was one > thing. > Your problem is that you flaunted the fact that YOU ARE IGNORANT OF > THE > RELEVANT DIFFERENCES BETWEEN THE TWO. > > ZFC is a set theory. PA is a number theory. > PA doesn't know what an infinite set is. ZFC does. > That is the main reason why ZFC can prove that PA is consistent > (a model of PA *has* to be infinite, and PA can't prove that anything > is infinite, since in its standard model, NOTHING IS).
From: Charlie-Boo on 26 Jun 2010 23:18 On Jun 25, 2:16 pm, Chris Menzel <cmen...(a)remove-this.tamu.edu> wrote: > On Thu, 24 Jun 2010 21:21:05 -0700 (PDT), Charlie-Boo > <shymath...(a)gmail.com> said: > > > 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, > > No it isn't. (Curious that you continue to assert otherwise.) What does that assertion mean, actually? > > 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? > > It is an elementary exercise. A proof can be found in almost any > reasonably thorough text in set theory. You know, you've always been a perfect target. You are one of those who try being condescending AND rely on the "vague = undefined = unrefutable" principle, while I am one who loves to define the undefined and thus refute the unrefutable (see past posts.) Ok, so if one went through a bunch of well-referenced texts, the great majority of them would contain a proof of PA written in ZFC - or however you think it should be phrased? And being elementary, it must be small and completely given? And being elementary, it should be in almost all texts. How about one that is on-line (from all those many instances of its being published)? Or a couple of your favs? C-B
From: George Greene on 27 Jun 2010 00:21 On Jun 25, 5:14 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > So why not have a try at it? You'll find all the details you need in any > decent text. This is a little sparse. I really don't think that the model existence theorem is going to leap out at him here.
From: Charlie-Boo on 27 Jun 2010 01:02 On Jun 27, 12:21 am, George Greene <gree...(a)email.unc.edu> wrote: > On Jun 25, 5:14 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > So why not have a try at it? You'll find all the details you need in any > > decent text. > > This is a little sparse. > I really don't think that the model existence theorem is going to leap > out at him here. No no no. You have to prove it using ZFC's axioms and rules only. C-B
From: Transfer Principle on 27 Jun 2010 01:54
On Jun 26, 7:41 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > "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. And the month of June continues with more and more posters coming out of the woodwork to challenge ZFC. This thread marks the return of Charlie-Boo and Srinivasan. Surprisingly, Charlie-Boo is one ZFC challenger whom I have yet to defend, even though I know that he's been posting here for years. I still remember several years back when he once compared ZFC to noodle soup. (Of course, I don't know whether Charlie-Boo still considers ZFC to be like soup anymore.) In this thread, Charlie-Boo is criticized for lumping together ZFC/PA as if they were interchangeable. I must point out that I myself lump them together all the time, but not because I consider them interchangeable -- we know that ZFC is a much stronger theory than PA. (Okay, okay, I mean that _a_suitable_extension_of_ZFC_ is a much stronger theory than PA....) Nonetheless, I lump ZFC and PA together as the two main standard _theories_ (not "theorists"). In particular, I often make statements such as, "Those who use standard theories such as ZFC/PA are much less likely to be called five-letter insults than those who use other theories." In this case, I'm not saying that ZFC and PA are equivalent to each other, but only that either theory is a suitable theory to use if one wants to avoid five-letter insults. Srinivasan, meanwhile, is trying to come up with NAFL, which is supposed to be an alternative _logic_ to FOL. If I remember correctly, in NAFL, it's possible for some statement to be similarly true _and_ false, unlike in FOL. Srinivasan was once fascinated by Ed Nelson's set theory, called Internal Set Theory or IST. One axiom schema of IST is called the Transfer Principle. I came up with my current username right in the middle of a discussion about IST with Srinivasan. Both Srinivasan and IST's creator appear to be sympathetic to finitism. Srivinasan discusses ZF-Infinity+~Infinity, a theory which can be used by finitists, while Nelson is working on a proof that PA is inconsistent. And of course, if Nelson's proof goes through, it would also prove that ZFC is inconsistent, since, as so many were quick to tell Charlie-Boo, ZFC proves that PA is consistent. Hughes will undoubtedly disagree with me, but I find the arrival of all these opponents of ZFC at the same time simply hilarious... |