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 28 Jun 2010 20:05 On Jun 28, 5:48 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > I believe that Srinivasan is on his way to giving a theory that will satisfy most finitists. WHAT theory? At this point, he's proposing some kind of cut back of the language of set theory, which would leave the language of equality theory. Or, I GUESS, he has in mind some "rule" or something that disallows formulas from "addressing" infinite sets. Yeah, let me know when we get a formulation of THAT so that there is a decision procedure to determine whether a given string is or is not a formula. MoeBlee
From: Chris Menzel on 28 Jun 2010 21:09 On Mon, 28 Jun 2010 05:02:20 -0700 (PDT), Charlie-Boo <shymathguy(a)gmail.com> said: > On Jun 28, 3:39 am, Tim Little <t...(a)little-possums.net> wrote: >> On 2010-06-27, Charlie-Boo <shymath...(a)gmail.com> wrote: >> >> > PA is not used but ZFC is? But ZFC invokes the Peano Axioms carte >> > blanche to represent N >> >> ZFC does not invoke the Peano Axioms at all. In fact to say that a >> formal theory "invokes" anything is at the very least somewhat odd. > > How do you prove Peano's Axioms in ZFC? A ZFC axiom gives them. False. Exactly why it is false has been explained to you numerous times, but you have no interest in understanding. Or, perversely, like a Tea Bagger politician, you are more interested in saying crazy stuff for the sake of attention (in this quirky and insignificant little corner of the net) than the facts.
From: Charlie-Boo on 28 Jun 2010 22:07 On Jun 28, 12:24 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 28, 7:19 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > Which post by MoeBlee says anything about how to prove PA consistent > > in ZFC? > > About a couple of years ago, I posted to you about it. As to this > thread, like Alan, I recall recently posting on it but can't find the > post in this thread (mabye it was in another recent thread?). > > In any case, you wrote to Aatu something to the effect that ZFC (by > the way, Z-R is more than adequate) provides for N and then proves the > axioms of PA about N (or something like that; I'm not representing > that I'm accurately capturing your remark; but merely trying to remind > you of the post). > > And that is essentially how Z-R proves the consistency of PA: > > In Z-R we define '0' (the empty set), 'w' (omega), 'S' (successor > operation for omega), '+' (addition operation for omega), > '*' (multiplication operation for omega). Then we prove, as theorems > of Z-R, the axioms of PA. Thus <w 0 S + *> is a model of PA. > > And it's easy enough to see that if a theory has a model then that > theory is consistent. Is that an axiomatic proof in ZFC? 3. If a theory has a model then that theory is consistent. Rule of Inference: It's easy enough to see. Ha! C-B > And, of course, since Z-R is a subtheory of ZFC, perforce, anything Z- > R proves also ZFC proves. > > MoeBlee
From: Charlie-Boo on 28 Jun 2010 22:21 On Jun 28, 7:06 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 28, 6:38 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > > > > > On Jun 27, 2:29 am, Transfer Principle <lwal...(a)lausd.net> wrote: > > > What's CBL? Is it "Charlie-Boo logic?" If so, then I'd like to > > > learn more about this challenger to FOL. > > It's not really a challenger to FOL. In fact, it uses FOL for what it > > does well: represent functions and relations over a single universal > > set - or subsets of it. > > CBL replaces FOL in certain contexts where we don't have a single > > universal set, because the domains of the components of functions and > > relations are not only different, they have diferent cardinalities! > > This is why the wffs for ZFC are so long, complex and sometimes > > debated as to validity. The primitives of something so primitive as > > set theory should be able to please Occam. And they can - in CBL. > > The ZFC axioms can be stated in a fraction of the size as using FOL. > > And how is the Theory of Computation formalized? How do we express > > its fundamental theorems and proofs? In CBL it is just a wff with a > > particular value of Q. > > Interesting. Thanks for the explanation. > > > Where does that occur? That is the definition of metamathematics. We > > are trying to draw relationships between sets of differing cardinality > > - e.g. to equate the aleph-1 set of wffs > > Are there really aleph-1 wffs? In the MoeBlee-Srivinasan subthread, we > have MoeBlee explaining that PA is an infinite theory: > > MoeBlee: > In particular, we may specify a certain > set of symbols and arity function so that that system is a language > for a first order theory such as PA, then specify PA to be the theory > that is the closure of the INFINITE set of axioms (the induction > schema is an infinite set of axioms) that we specify. > > but MoeBlee doesn't state whether this "infinite" is countable, > aleph_1, or larger. (Similarly ZFC is also infinitely axiomatized, so > we know that it's infinite.) I could've sworn that someone posted > that there are only countably many wff's (in the language of either > PA or ZFC) since each wff is a finite formula, each taken from a > countable set of symbols (which includes the countably many > variables, as also mentioned by MoeBlee). > > > with the aleph-2 set of functions over wffs. > > If by "functions over wffs," Charlie-Boo means functions from the > set of wffs to itself, then there are aleph_0^aleph_0 = continuum > many such functions. Of course, if the metatheory is > ZFC+"c=aleph_2," there could be aleph_2 many such functions. > > > In metamathematics, we want to know if a certain set can be > > represented within another set by substituting a constant for one > > component. We may also want to know and use this constant. So we > > relate a wff to a set. The universal sets having different > > cardinalities, we need a relation over sets of different > > cardinalities. > > OK, I somewhat see what this is leading to. Charlie-Boo does > mention "universal sets" such as V (a proper class in NBG, > but apparently a set in CBL), so that one can replace the wff > such as "x=x" with "xeV." It's the proofs that are amazingly short - although the wffs are short as well. "xeV" is not a CBL wff. See what I wrote. M#P/Q is a fundamental operation over object M and relations P and Q, meaning P=Q(M). P/Q is defined in terms of that: (existsM)M#P/Q. The rest is standard Logic although for Recursion Theory I introduce a symbol for "functionally equivalent" and define it in terms of CBL expressions, as it is used often in that branch of Computer Science and makes the expessions more natural. Tell me your favorite metamathematical theorem - one for which you know a proof well. I will tell you how CBL handles theorems of that type. C-B > This isn't much shorter, but I assume > that wffs such as "x is an ordinal" (a bit lengthy when expanded > to full primitives) would become something like "xeOn" in CBL. > > Am I on the right track here?- Hide quoted text - > > - Show quoted text -
From: Chris Menzel on 28 Jun 2010 22:17
On Mon, 28 Jun 2010 04:56:30 -0700 (PDT), Charlie-Boo <shymathguy(a)gmail.com> said: > On Jun 27, 6:31 pm, Chris Menzel <cmen...(a)remove-this.tamu.edu> wrote: >> On Sun, 27 Jun 2010 10:50:16 -0700 (PDT), Charlie-Boo >> <shymath...(a)gmail.com> said: >> >> > ... >> > ZFC declares that there is a set that satisfies Peano's Axioms and >> > defines N to be that set. >> >> You have it completely backwards. The set is defined first, independent >> of any mention of PA. > > So? So what you said was false. You said that ZFC defines N to be a set that satisfies the Peano Axioms. False. > Doesn't that set satisfy Peano's Axioms? No. Axioms are true in models and the set in question (assuming you mean the set of finite von Neumann ordinals) is not itself a model. Rather it is the domain of a model in which the axioms are true. > Whether it says it or not doesn't change the question. What question? >> One can then prove (in ZF) that that set (and indeed, infinitely many >> others) can serve as the domain of a model of PA. > > And then doesn't it define N to be that set? No. Again, a certain set is defined in ZF in purely set theoretic terms, e.g., the smallest set N containing the empty set and such that, if s is in N, so is sU{s}. A model can then be defined in ZF that interprets the primitives of the language of arithmetic in terms of N. (To do this all in pure ZF, constants, predicates, sentences etc are coded as sets.) It can then be shown (in ZF) that all of the Peano axioms are true in this model. So, to complete the answer to your question: N is never *defined* to be a set -- or, more exactly, the domain of a model -- in which the Peano axioms are true. That *turns out* to be the case after the set and corresponding model are defined. I realize none of this gets through your obdurate skull, but hopefully an interested reader or two might benefit. |