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 1 Jul 2010 19:55 On Jul 1, 6:26 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > Or was this one of those cases where nobody really does that because > it's too messy, or because it's so easy, one of those "Goldilocks" > problems? Gotta go. Life too short for fruitlessly trying to communicate with you. I figured this stuff out pretty much by reading the books and going a step further to formalize every detail I needed so that I could see that complete formalization is possible and to see what it would be, so you should be able to do it to. I'd recommend starting at square one, with what I think is the best book on learning to work in the first order calculus: "Logic: Techniques Of Formal Reasoning" by Kalish, Monatague, and Mar. My very best practical answer to address the root of your confusions is for you to complete that book so that you know it throroughly. THEN we might be able to talk about even more technical matters. MoeBlee
From: herbzet on 1 Jul 2010 20:52 MoeBlee wrote: > > CORRECTION below: > > On Jul 1, 5:10 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > However, even though the definition I gave is okay, and provides these > > theorems, > > > > a theory T is inconsistent <-> T proves a contradiction > > > > a set of axioms X for a theory T is inconsistent <-> X proves a > > contradiction > > Oops, delete that. Not sure if "that" is the previous sentence, or the previous 2 sentences. > > a theory T that is axiomatized by X is inconsistent <-> X proves a > > contradiction, > > > > I realize that in my own notes I actually do use a different > > definition, which, again, is equivalent for THEORIES but different for > > arbitrary sets of formulas: Missing the nuance here, unless you are distinguishing between a set Gamma of formulas *implying* both P and ~P, and a set Gamma of formulas *containing* both P and ~P. But let that go ... > > a set of formulas G is consistent <-> there is no formula P such that > > P and ~P are provable from G OK, so far so good. > > a set of formulas G is inconsistent <-> G is not consistent Can't object to that. > and I think that is an easier and nicer definition to work with as it > gives all at once: > > a set of formulas G is inconsistent <-> G proves a contradiction > > a set of formulas G is inconsistent <-> G proves every formula (in the > language) No doubt missing some of the issues here, but: For classical logic the following sets are extensionally equivalent: 1) a set of formulas G that proves both P and ~P for some formula P. 2) a set of formulas G that proves a contradiction, i.e., some formula (P & ~P). 3) a set of formulas G that proves every formula (in the language). For non-classical logics, these may diverge. What (3) describes is called a trivial theory. We don't necessarily care so much about inconsistency -- a paraconsistent logic will tolerate some inconsistency of type (1) or (2) in a theory T without T's necessarily becoming trivial. We don't worry so much about inconsistency per se -- we worry about triviality instead. Tra-la-la! -- hz
From: herbzet on 1 Jul 2010 20:55 herbzet wrote: > For classical logic the following sets are extensionally equivalent: Well, that's not exactly right, but you know what I'm getting at. -- hz
From: K_h on 1 Jul 2010 22:06 "R. Srinivasan" <sradhakr(a)in.ibm.com> wrote in message news:90ce086b-c089-4b90-866d-c7391e95bbd6(a)j8g2000yqd.googlegroups.com... On Jul 1, 3:15 am, "K_h" <KHol...(a)SX729.com> wrote: > "R. Srinivasan" <sradh...(a)in.ibm.com> wrote in message > > news:46d58d89-34b1-40a9-a5a8-1ee250ba57e3(a)e5g2000yqn.googlegroups.com... > On Jun 29, 8:33 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 29, 2:09 am, "R. Srinivasan" <sradh...(a)in.ibm.com> > > wrote: > >> > > ZF-"Inf'+"~Inf" >> >> > > That theory entails that every object is finite. And >> > > there is no >> > > definition of any infinite object possible in that >> > > theory. >> >> > OK. Here I want ~Inf to be stated in the form that you >> > mentioned, that is, every set is hereditarily finite. >> >> Why do you think the axiom of infinity is false? What is >> the basis for your belief in ~Inf? To me it is >> self-evident that all the naturals exist. >> > First of all I happen to work in a logic (NAFL) where I have a *proof* > of ~Inf. Essentially, if you define truth (as provability) such that > all vestiges of Platonism are thrown out, infinite sets will not There are several problems with this. First, you have not explained how or why you think that mathematical Platonism is false. Second, allowing N to exist as a proper class is platonic (see my remarks below). Third, Cantor's theorem suggests that N should be a set since |N|<|P(N)| cannot hold true for a proper class. Fourth, there is no reason why truth should be defined as provability, especially in light of the incompleteness theorems. > Yet if > infinitely many natural numbers are exceeded *within* N, it seems that > the only way out is that N must contain an infinitely large number. Suppose that were the case then it would still be possible to get what is usually called N by putting just the finite members into their own set. > This is precisely the intuition that leads to nonstandard models of > arithmetic, where there are nonstandard integers that exceed every > "standard" natural. To call such numbers "finite" is grotesque, to say > the least. Yet that is the only way to save the consistency of > classical Peano Arithmetic. We have to sacrifice our well-known and > well-accepted intuition of what "finite" means, which is something I > am not willing to do. No, the consistency of PA is not in any danger here. In the non-standard models "finite" means something different. To retain the usual meaning of "finite" we need to go beyond first-order logic to second-order logic. Once that is done then it is possible to distinguish between standard and non-standard models and PA is consistent in all of them. > If the above considerations do not already leave a bad taste in the > mouth, consider the definition of N as a set. It is an essentially > impredicative definition. Here I am talking about the simple basic > definition of N, which uses universal quantifiers in an essential way. > These quantifiers quantify over an universe that already contain N. > That such a definition is "harmless" is a commonly stated assertion. > If you think carefully, such a defense of circularity is based on > Platonism, namely, that N "really" exists, and our attempted > definition only tries to access something that is already "out there" > in the universe of sets. Note that we do not have this problem with > finite sets, even if these are defined using quantifiers. Because we > can always define them predicatively by listing their elements. But you have already claimed that N really exists out there as a proper class. And so, by your own logic, you have engaged in circular reasoning. The insight underlying the diagonal argument shows that |N|<|P(N)| holds true for N and so N cannot be a proper class. Therefore, inductive sets are needed to make sense of things like square roots. There are many algorithms that calculate square roots, for example root(2). So the founding axioms should guarantee a set, N, that makes possible a list of every numeral in root (2): 0 <--> 1 1 <--> . 2 <--> 4 3 <--> 1 4 <--> 4 5 <--> 2 6 <--> 1 7 <--> 3 8 <--> 5 9 <--> 6 .... Unless you deny the existence of root(2) and deny the insight underlying the diagonal argument, inductive sets have to really exists out there in the universe of sets. That, together with the existence of algorithms that enumerate all digits of root(2), provides a rock solid justification for an axiom stating the existence of inductive sets. > Here is a post (by Brian Hart) in the FOM newsgroup that says > Platonism is essential to defend the impredicative methods used in > modern logic, physics, mathematics: > > http://www.cs.nyu.edu/pipermail/fom/2010-May/014713.html > > \begin{quote} > If one axiomatizes the logical universe (the one containing strictly > logical objects such as proper and hyper-classes) impredicativity is a > requirement as these objects cannot be defined non-circularly. > \end{quote} So when you write "However, there can and do exist infinite classes, like N, the class of all natural numbers" you are being platonic. There is nothing wrong with that unless you are denying Platonism. _
From: Chris Menzel on 2 Jul 2010 05:31
On Thu, 01 Jul 2010 17:16:35 +0100, Frederick Williams <frederick.williams2(a)tesco.net> said: > Charlie-Boo wrote: >> >> The next thing to actually do is to show how or why PA can't prove PA >> consistent in detail > > Hilbert & Bernays do that. Peter Smith's wonderfully lucid book _An Introduction to Gödel's Theorems_ highly recommended for this. |