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 08:43 On Jun 27, 2:29 am, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 26, 6:09 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > On Jun 24, 6:04 pm, George Greene <gree...(a)email.unc.edu> wrote: > > > ZFC is one thing. PA is another. > > And CBL is still another. However, CBL proves theorems with proofs > > that are about 1% the size of those published, while ZFC and PA take > > about 10 times the size published. So which is best? > > What's CBL? Is it "Charlie-Boo logic?" If so, then I'd like to > learn more about this challenger to FOL. Just remembered another improvement to FOL - and this is a big (significant) one: 4. Bounded quantifiers are a big kludge (doesn't fit in and is added for one case rather than the general case - lots of programming languages do this - then later try to fix it.) The interest is that (allA<N)P(A,x) is recursive when P is, while (aA)P(A,x) is not recursive or even r.e. in general. So this kludge syntax and its rule are added. But we can prove that using the existing FOL syntax and some well- chosen rules of inference that have wide applicability (e.g. formalizing Rosser 1936 at a detailed level.) The expression is (aA) [LT(A,N)=>P(A,x)] where LT(a,b) means a<b. How do we prove it recursive given that P is? I mean formally with axioms and rules (everything I do is.) Another puzzle related to this: Give a function of x that always produces a set containing a finite number of elements but the relation between x and the elements of that x ("y is in the function of x") is not recursive. C-B
From: Charlie-Boo on 29 Jun 2010 08:58 On Jun 29, 8:23 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > What of ZFC's set theoretic axioms is necessary - especially not > > bookkeeping ones like sets existing that are used throughout PA and > > are not needed in every proof? > > Various bookkeeping axioms about set existence are used in the proof. Reference? You know, I asked for nonbookkeeping. If it's bookkeeping (not needed in PA), then the proof can be carried out in PA. Uh-Oh! > Of course, for the consistency of PA ZFC is a huge overkill. No excuses or obfuscations, plz. (I told you you've started acting like the stupid fucks.) Where's the beef (ZFC proof)? C-B > We can prove > PA is consistent in e.g. ACA, using a (definable) truth predicate for > arithmetical statements. > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, dar ber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Charlie-Boo on 29 Jun 2010 09:06 On Jun 29, 8:28 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > George Greene <gree...(a)email.unc.edu> writes: > > I really don't think that the model existence theorem is going to leap > > out at him here. > > Pretty much any elementary text contains an account of the set theoretic > construction of the various number systems. It's a trivial exercise to > verify the axioms of PA hold in the structure of naturals. Combined with > the soundness of first-order logic this immediately yields the > consistency of PA. Can that be carried out in ZFC? Who has? I've asked like 5-10 times. The only answers I've gotten were YES then changing the subject or a BS reference (for $400.) Elementary, trivial? See - more fuckhead nonsense (condescending.) A: If it's trivial then give it. DUH: Mathematicians don't give trivial proofs. A: When's the last time you read a math book? 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: Charlie-Boo on 29 Jun 2010 09:15 On Jun 29, 8:30 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > herbzet <herb...(a)gmail.com> writes: > > "Jesse F. Hughes" wrote: > > >> But, Walker, you really have the wrong impression of me. I come to > >> sci.math mostly to read the cranks. I'm not proud of that fact > > > *I* am proud of you, that you would make this startling announcement. I say the cranks are the ones who give BS references, make unsubstantiated statemements and who practice arguing ad hominem. Agreed? Or are you so base that you won't admit to even principles of morality? C-B > Coincidentally, I am reading /Idiot America/ by Charlie Pierce. The > book has much to do with cranks, though not of the mathematical sort so > much as the political and other sorts. There's a vaguely relevant > passage. > > The value of the crank is in the effort that it takes either to refute > what the crank is saying, or to assimilate it into the mainstream. In > either case, political and cultural imaginations expand. Intellectual > horizons expand. > > Now, contrary to Walker's fantasy, none of the cranks here have > anything worth "assimilating" into mathematics, of course, but there is > something to be said with the effort required to refute a bad argument. > Of course, no one here really thinks that any argument will effectively > end a crank's pursuit of nonsense, but that is nonetheless the > intellectually interesting bit. > > Of course, I'm really here for lower entertainment. I want posts about > the Hammer, about how surrogate factoring moves the stock market, about > the most influential mathematicians on the planet. But still I pretend > to care about arguments, if only for appearance's sake. > > -- > Jesse F. Hughes > > "Most of my research is irreducibly complex." > -- James S. Harris
From: Charlie-Boo on 29 Jun 2010 09:17
On Jun 29, 8:34 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > 1. Is there a proof in ZF-Inf+~Inf that every set is finite? > > It appears I once posted a proof. It relies, in an essential way, if I > recall correctly, on the axiom of replacement. You recall incorrectly. It actually consisted of drawing of little girls playing outside your window. C-B > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, dar ber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |