How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: herbzet on 1 Jul 2010 16:28 "R. Srinivasan" wrote: > 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 > survive. However, there can and do exist infinite classes, like N, the > class of all natural numbers. But quantification over classes is not > allowed and classes can only be defined by construction -- there is no > "arbitrary" infinite class in NAFL theories. Despite these seemingly > severe restrictions, I show that it is possible to define a method for > real analysis in NAFL based on translating Euclidean geometry into a > theory of finite sets with classes. I also show that the paradoxes of > classical real analysis, like Zeno's paradox, Banach-Tarski paradox, > etc. will be eliminated in such a system of real analysis. > > On a more intuitive level, how can we fault the existence of the > infinite set N? If you consider the statement "All natural numbers are > not upper bounds for N" and ask "how many natural numbers are exceeded > by some element of N?", the answer has to be "infinitely many". 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. > 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. > > 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. > > 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} > > This post seemed to be the first sensible one in an FOM thread where > dozens of badly-off-the-mark posts had appeared earlier. Guess what? > Precisely after this post appeared, the "moderator" of FOM, Martin > Davis, decided to call off the discussion: > > http://www.cs.nyu.edu/pipermail/fom/2010-May/014716.html > > \begin{quote} > This discussion has long since reached the point of diminishing > returns. Hereafter only messages on this topic judged to be of very > special interest will be posted. > \end{quote} > > FOM is supposed to be the newsgroup where all the elite logicians and > philosophers ponder the foundations of mathematics and logic. And you > can see the narrow-minded intolerance that prevails at that level just > by looking at this thread and the manner in which it was closed, > without allowing any reply to the post of Brian Hart I quoted above. Hi, RS. Haven't been following this current thread, but I asked you a long time ago if NAFL assumes classical propositional logic. Do you know what propositional logic is assumed in NAFL? -- hz
From: MoeBlee on 1 Jul 2010 18:10 On Jul 1, 3:48 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > So, according to _your_ "precise" definition, S = {GC, cGC} is precisely > a consistent set of formulas, right? I resisted this time going down another suckhole of yours. I have nothing to say about your cGC whatever it is. 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 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: a set of formulas G is consistent <-> there is no formula P such that P and ~P are provable from G a set of formulas G is inconsistent <-> G is not consistent MoeBlee
From: MoeBlee on 1 Jul 2010 18:15 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. > 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: > > a set of formulas G is consistent <-> there is no formula P such that > P and ~P are provable from G > > a set of formulas G is inconsistent <-> G is not consistent 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) and a theory is a special case of a set of formulas, as is an axiomatization of a theory. MoeBlee
From: Charlie-Boo on 1 Jul 2010 19:23 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? C-B > > and sets in ZFC. That's why it is > > expressed differently - different alphabets as well. > > -- > I can't go on, I'll go on.
From: Charlie-Boo on 1 Jul 2010 19:26
On Jul 1, 4:09 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jul 1, 2:30 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > The point is that ZFC would have to have an axiom other than Infinity > > (i.e. PA) that is necessary to prove PA consistent in order for it to > > be impossible in PA and possible in ZFC. But again you don't show > > that. > > Talking to myself: PLEASE MoeBlee don't waste a second more of your > time trying to get through to the ineducable Charlie-Boo. Listen, > MoeBlee, no matter how much you explain, no matter how detailed or > general, Just the title of a reference with a proof of the consistency of PA carried out in ZFC would do. 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? C-B > Charlie-Boo will still persist with yet more of his > confusions over even the most simple matters such as the difference > between Z set theory and first order PA. > > MoeBee |