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 16:25 Ki Song wrote: > > Perhaps an analogy is in order! > > I feel like what Charlie-Boo is asking people to do is analogous to > asking someone to perform the addition: > > Sqrt{2}+Sqrt{3}, with a decimal precision of 10^(10^(10^100)) place. I'm getting to appreciate the use of a good analogy in a debate. I find it hard to come up with them. Well done. -- hz
From: herbzet on 1 Jul 2010 16:26 Charlie-Boo wrote: > You see, if you lie with dogs, you are a dog. And I know exactly how > dogs bark. ... since I am a dog, beware my fangs ... - Merchant of Venice III,3 -
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 |