How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: George Greene on 27 Jun 2010 13:33 On Jun 27, 1:02 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > No no no. You have to prove it using ZFC's axioms and rules only. Shut UP, fool! You DON'T GET to tell ME how this works! I have two degrees in logic AND YOU DON'T! And in any case, I WAS TALKING ABOUT proving it "using ZFC's axioms and rules only", or at least as close to "only" AS YOU CAN GET, GIVEN that ZFC, contrary to your misguided assertion, DOES NOT "include" PA! PA has a zero, a successor function, an addition function, and a multiplication function (as well as associativity and induction). ZFC *DOES*NOT*HAVE*ANY* of those things! However, "using ZFC's axioms and rules only", you can PROVE THE EXISTENCE OF SETS (and sets of sets, and sets of ordered pairs, i.e., FUNCTIONS) that WILL have these properties, if you set them up in appropriate relationships with each other. Then, since ZFC is not going to, by itself, NAME these things "zero", "successor", "addition", "multiplication", "associativity", or "induction", YOU CAN INTERPRET these concepts from PA *as* these ZFC sets, SIMPLY BY SAYING what "stands for" what.
From: George Greene on 27 Jun 2010 13:36 On Jun 27, 1:38 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > If you're saying by providing a model, then you are wrong. Your being ignorant does not make me wrong. It's just a fact about first-order theories that consistent ones have a model and inconsistent ones don't. Model-existence is equivalent to consistency for this class of theory, so proving model-existence is (one way of) proving consistency. There are other ways, but just because you would prefer some other way DOES NOT invalidate THAT way. In EITHER case, INTERPRETATION remains CRUCIALLY relevant, because nothing in ZFC "already/automatically" *IS* ANY concept from PA.
From: George Greene on 27 Jun 2010 13:37 On Jun 27, 1:54 am, Transfer Principle <lwal...(a)lausd.net> wrote: > Hughes will undoubtedly disagree with me, but I find the > arrival of all these opponents of ZFC at the same time > simply hilarious... You're an idiot. These opponents ARE NOT all arriving at the same time. Srinivasan was talking about NAFL here 4 years ago. Herc has been around longer than that, and so has Nam.
From: Charlie-Boo on 27 Jun 2010 13:50 On Jun 27, 1:22 pm, William Hale <h...(a)tulane.edu> wrote: > In article > <dbebec73-757b-4de5-ae60-d5f6a6ab8...(a)t10g2000yqg.googlegroups.com>, > > > > > > Charlie-Boo <shymath...(a)gmail.com> wrote: > > On Jun 27, 5:18 am, William Hale <h...(a)tulane.edu> wrote: > > > In article > > > <ff54cc7d-b23f-4a45-9040-0459145ff...(a)j8g2000yqd.googlegroups.com>, Charlie- > > > Boo <shymath...(a)gmail.com> wrote: > > > > [cut] > > > > > If ZFC can't calculate what PA can, how can anyone say that ZFC is a > > > > good basis for doing mathematics - PA is used by lots of > > > > mathematicians. > > > > 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. > > > PA is not used but ZFC is? But ZFC invokes the Peano Axioms carte > > blanche to represent N - so PA is used by ZFC and thus by all of these > > Mathematicians. > > ZFC does not invoke the Peano Axioms to represent N. Textbooks may ZFC declares that there is a set that satisfies Peanos Axioms and defines N to be that set. Whether it uses the phrase Peanos Axioms or not, those are the axioms that are being listed and used. C-B > mention the Peano Axioms when they show how ZFC incorporates what it > deems to be the natural numbers, but this mention of PA is only for > putting things in a historical perspective or to give some motivation > for what is going on in ZFC, but it is not necessary for ZFC to have any > mention of PA in order to develop natural numbers. > > A textbook could mention how Euclid developed natural numbers (for > historical purposes or to show similarities etc), but this does not mean > that ZFC invokes the five Euclidean postulates of geometry. > > > > > Good point! > > > C-B- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text -
From: Charlie-Boo on 27 Jun 2010 13:56
On Jun 27, 1:26 pm, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > Charlie-Boo wrote: > > Gentzen's consistency proof "reduces" the consistency of mathematics, > > not to something that could be proved. Wikipedia > >http://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof > > > Wiki doesn t say anything about ZF in its write-up of Gentzen s proof > > of the consistency of PA! What happened?? > > I cannot speak for the Wikipedists, but can you not see that the proof > could be formalized in ZFC? The question was whether anyone had proven PA consistent using ZFC. You said they had and said Gentzen did. But Wikipedia never mentions ZF or ZFC in its description of Gentzen's proof. So the question remains, did Gentzen really prove PA consistent using ZFC, or is this yet again another example of good 'ole bullshit? Where did you see that he used ZFC in his proof? C-B > I don't know what to make of > > It "reduces" the consistency of a simplified part of mathematics, > not to something that could be proved (which would contradict the > basic results of Kurt G del), but to clarified logical principles.. > > But then that's Wikipedia for you. > > -- > I can't go on, I'll go on. |