How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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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.
From: Chris Menzel on 2 Jul 2010 05:34 On Thu, 1 Jul 2010 08:41:44 -0700 (PDT), Charlie-Boo <shymathguy(a)gmail.com> said: > On Jul 1, 2:04 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> Charlie-Boo <shymath...(a)gmail.com> writes: >> > To see it means to have actually created it, and its actual creation >> > would answer the very interesting question of whether ZFC can prove >> > that PA is consistent even though PA can't. >> >> We already know the answer. The axioms used in the proof are a >> restricted form of comprehension, infinity, union and pairing. > > Why do we need each of these? Of course infinity = PA = arithmetic. > But why are the others needed? These are the questions you could answer for yourself with just a bit of study. The tragic thing is, these topics are so marvelously interesting and yet you insist on dwelling in ignorance. (Infinity = PA? So sad...)
From: Chris Menzel on 2 Jul 2010 05:40 On Wed, 30 Jun 2010 21:23:23 -0700 (PDT), Charlie-Boo <shymathguy(a)gmail.com> said: > On Jun 29, 9:51 pm, herbzet <herb...(a)gmail.com> wrote: >> "Jesse F. Hughes" wrote: >> > Of course, I'm really here for lower entertainment. I want posts about >> > the Hammer, about how surrogate factoring moves the stock market, about >> > the most influential mathematicians on the planet. But still I pretend >> > to care about arguments, if only for appearance's sake. >> >> Well, I didn't come here for the low comedy -- that's simply not an idea >> that had occurred to me. But, it being made explicitly an option -- I >> suppose it's a valid choice, even one of some value. >> >> If I have any reservation, it would just be that I don't think that >> encouraging crankery for its entertainment value > > crank: "an unbalanced person who is overzealous in the advocacy of a > private cause" You're talking about you and CBL, right?
From: Frederick Williams on 2 Jul 2010 10:39
Charlie-Boo wrote: > > On Jul 1, 3:50 pm, Frederick Williams <frederick.willia...(a)tesco.net> > wrote: > > Charlie-Boo wrote: > > > No, PA and ZFC both have Peano's Axioms. > > > > What formulations of first order PA and ZFC have _any_ of their > > non-logical axioms in common? None. Why? Well, for starters, all of > > the proper axioms of ZFC have the predicate symbol $\in$ in them, none > > of PA's axioms do because $\in$ is not even in the language of PA > > > > > The only difference is the > > > universal set, which is N in PA > > > > There are no sets, universal or otherwise, in first order PA. > > What does (existsX)P(X) mean? It means that there's a natural number which has property P. -- I can't go on, I'll go on. |