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 18:40 By the way, here's Aatu's proof: Theorem (of Z-"ax regularity"-"ax infinity"+"~ ax infinity"): Ax x is finite. PROOF ['P' stand for 'power set'] Let 'F' be a 1-place operation symbol defined as follows: F(y) = the n such that ((y is finite & n is a natural number & y is equnumerous with n) or (y is infinite & n=0)). Toward a contradiction, suppose x is infinite. So {F(y) | y in Px} is the set of natural numbers, which is successor inductive. MoeBlee
From: MoeBlee on 28 Jun 2010 18:46 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? Aatu, Daryl, or anyone help me out? MoeBlee
From: Transfer Principle on 28 Jun 2010 18:48 On Jun 28, 9:14 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 27, 10:49 pm, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote: > > Here is one possible approach. Consider the theory F that I had > > defined in another post, where > > F = ZF - Inf + D=0, > > Where 0 is the null set and D is defined as > > D = {x: An(x not in P_n(0))} I wonder why Srinivasan can't simply use the nth level of the cumulative hierarchy, V_n, so that we don't have to worry about whether P_n(0) is welldefined or not. > Prove (and please state the formal language, logic, and axioms in > which you conduct this proof) that there exists a D such that for all > x, we have x in D iff An x not in P_n(0). > Unless you can do that, then your 'D' is not well defined. OK, I see the problem here. In standard ZF, the object that Srinivasan calls "D" would be too large to be a set. Thus, it's not evident that the set D exists. Rather than attempt to prove it, why can't we just let this be the axiom itself -- so instead of writing D=0, the axiom becomes: ED (Ax (xeD <-> An (~xeV_n))) exactly the statement that MoeBlee asks Srinivasan to prove (except inserting V_n). We can worry about whether such a set can equal 0 down the road. > > Here P_n(0) is power set operation iterated n times on 0, and P_0(0) = > > P(0), P_1(0)=P(P(0)), etc. Clearly only hereditarily finite sets can > > exist in models of F. > Prove it. OK, I see the problem here. Even if F has a model in which every set is HF, there might be non-isomorphic models of F with sets which aren't HF at all. Still, I believe that Srinivasan is on his way to giving a theory that will satisfy most finitists.
From: Transfer Principle on 28 Jun 2010 19:06 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." 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?
From: MoeBlee on 28 Jun 2010 20:02
On Jun 28, 6:06 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > MoeBlee explaining that PA is an infinite theory: In the ordinary context, EVERY theory is infinite. Every theory is an infinite set of sentences. > MoeBlee doesn't state whether this "infinite" is countable, PA is countable. It's a countably infinite set of sentences. > Similarly ZFC is also infinitely axiomatized, so we know that it's infinite. Even theories that are FINITELY axiomatized are infinite. EVERY theory (in a context such as this) is infinite, since every theory is an infinite set of sentences. ZFC is countable. It is a countably infinite set of sentences. > 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). Right. MoeBlee |