How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: Charlie-Boo on 29 Jun 2010 06:34 On Jun 29, 12:18 am, Chris Menzel <cmen...(a)remove-this.tamu.edu> wrote: > On Mon, 28 Jun 2010 15:46:25 -0700 (PDT), MoeBlee <jazzm...(a)hotmail.com> > said: > > > One thing I don't know how to do is show the mutual-interpretability > > of PA and Y=ZF-"ax inf"+"~ax inf" > > > One direction seems not too difficult: interpreting PA in Y. > > > But how do we interpret Y in PA? Specifically, how do we define 'e' in > > PA and then prove, in PA, all the axioms of Y as interpreted in the > > language of PA? > > The best known approach uses a mapping that Ackermann defined from the > hereditarily finite sets into N There are too many sets to map them 1-to-1 with the natural numbers. C-B > that takes the empty set to 0 and, > recursively, {s_1,...s_i} to 2^(n_1) + ... + 2^(n_i), where n_i codes > s_i. For numbers n and m, let nEm iff the quotient of m/2^n is odd. > The relation E is obviously definable in PA. Ackermann showed that, by > defining the membership predicate as E, the axioms of Y are all theorems > of PA.
From: Charlie-Boo on 29 Jun 2010 06:57 On Jun 27, 9:40 pm, Chris Menzel <cmen...(a)remove-this.tamu.edu> wrote: > On Sun, 27 Jun 2010 17:14:02 -0700 (PDT), Charlie-Boo > <shymath...(a)gmail.com> said: > > >> > > ZF, for example, proves Con(PA) which, of course, PA does not > >> > > (assuming its consistency). > > > > > You have to assume PA is consistent? > > > > You have to ask? > > > Yes or no, please. > > Yes. (You *really* had to ask?) In Godel's proof that his system cannot prove its own consistency, he uses the caveat "assuming it is consistent". But he is not talking about PA at that point. He is talking about a system with a variable for the set of axioms, which includes consistent systems and inconsistent systems. Thus the "assuming it is consistent". Many unsophisticated people mistakenly refer to "assuming PA is consistent", not knowing that Godel was talking about sets of axioms outside of PA's. PA itself is consistent, as is easily proved based on the simple fact that the axioms and rules preserve truth. There is no "assuming PA is consistent", Chris Menzel - it is provably consistent. C-B
From: Charlie-Boo on 29 Jun 2010 08:03 On Jun 29, 6:56 am, master1729 <tommy1...(a)gmail.com> wrote: > CBL = ?? > > wffs = ?? > > sorry i just started reading this thread , maybe it has been explained already. # 1. CBL is NOT "Charlie-Boo's Language"! That is a common myth. It stands for Computationally Based Logic. It is a fairly simple extenstion to Logic that makes it easy to represnt relations between sets of differing cardinalities, which is central to metamathematics. Notice that Mathematical relations such as interger or real number addition relates sets of the same cardinality, in fact, the same set is the domain of each component. In metamathematics, we need to relates sets that are not only different, but even have a different cardinality. PR=the set of theorems, TW=true sentences, DIS=refutable sentences, HALT=(program,input) that halts, YES=that halts yes, SE=(set,elements) where the element is in the set. How do we say that program M enumerates set P? In CBL it is M#P/YES. This is a relation between natural numbers or strings (programs) and sets of numbers, sets with different cardinalities. M#P/TW means that M expresses P. M#P/SE means M is the set containing just what satisfies P. P/Q means (existsM)M#P/Q. So P/YES means P is recursively enumerable, P/TW means P is expessible, P/PR means P is representable, P/SE means P is a set, etc. With just the simple syntax of P/Q and M#P/Q we can represent all of the fundamental concepts of theoretical Computer Science. How are these notions represented in conventional Mathematics? By a variety of syntactic kludges or simply in English! If you are interested, I can next show you how to easily generate lots of proofs of very well-known theorems - proofs that are 1-2% the size of published proofs. C-B
From: Charlie-Boo on 29 Jun 2010 08:22 On Jun 29, 7:57 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > "R. Srinivasan" <sradh...(a)in.ibm.com> writes: > > The theory ZF-Inf+~Inf clearly proves ~Inf ("Infinite sets do not > > exist"). This proof obviously implies that "There does not exist a > > model for PA", for a model of PA must have an infinite set as its > > universe (according to the classical notion of consistency, which I am > > going to dispute shortly). Therefore we may take the proof of ~Inf in > > ZF-Inf+~Inf as a model-theoretic proof of the inconsistency of PA, > > which must be equivalent to its syntactic counterpart ~Con(PA). > > Without the axiom of infinity we can't prove that if a theory has no > model it is inconsistent. Indeed, we can prove in ZF-Inf+~Inf that some > consistent theories have no model. Still waiting for the published proof of PA consistency using only ZFC. Godel's 1st: Truth does not equal provability. C-B's 137th: Truth does not equal claimability. qed C-B > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, dar ber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 29 Jun 2010 08:37
William Hale <hale(a)tulane.edu> writes: > PA is not used by any mathematicians to do algebra, number theory, > real analysis, complex analysis, topology, or differential > geometry. These mathematicians represent most mathematicians. They use > ZFC as their axiomatic system. Most mathematicians don't use any formal theory in any apparent sense. Rather, they just go about their business of doing mathematics. That the mathematical principles and modes of reasoning they implicitly (or, rather less commonly, explicitly) invoke in this their business are formally captured in this or that formal theory is usually of no general mathematical interest. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |