How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
in [Math]
|
Prev: ? theoretically solved
Next: How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
From: Frederick Williams on 27 Jun 2010 13:26 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? 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.
From: Frederick Williams on 27 Jun 2010 13:29 Charlie-Boo wrote: > Hey Frederick, I bet you $449.94 that Gentzen's book doesn't contain a > proof that PA is consistent, carried out in ZFC. You on? No, but I do claim that it could be formalized in ZFC. > Or do you say things that you don't believe? Yes, but I don't think that's relevant here. -- I can't go on, I'll go on.
From: Charlie-Boo on 27 Jun 2010 13:31 On Jun 27, 1:21 pm, ah...(a)FreeNet.Carleton.CA (David Libert) wrote: > Transfer Principle (lwal...(a)lausd.net) writes: > > On Jun 26, 7:51=A0pm, Tim Little <t...(a)little-possums.net> wrote: > >> On 2010-06-26, R. Srinivasan <sradh...(a)in.ibm.com> wrote: > >> > The theory ZF-Inf+~Inf clearly proves ~Inf ("Infinite sets do not > >> > exist"). > >> Actually ~Inf does not assert "Infinite sets do not exist". =A0It only > >> asserts "there does not exist a successor-closed set containing the > >> empty set". > > > This has come up time and time again. I myself have claimed that > > the theory ZF-Infinity+~Infinity proves that every set is finite, > > and someone (usually MoeBlee or Rupert) points out that this > > theory only proves that there's no _successor-inductive_ set > > containing 0, not that there is no infinite set. > > > And every time this comes up, I want to say _fine_ -- so if > > ZF-Inf+~Inf _doesn't_ prove that every set is finite, then there > > should exist a model M of ZF-Inf+~Inf in which "there is an > > infinite set" is true, even though "there exists a set containing > > 0 that is successor-inductive" is clearly false (assuming, of > > course, that ZF is itself consistent), just as the fact that ZFC > > doesn't prove CH implies that there is a model of ZFC in which > > CH is false (once again, assuming that ZF is itself consistent). > > > Yet no one seems to accept the existence of this model M. > > > Either this model M exists, or ZF-Inf+~Inf really does prove that > > every set is finite. There are no other possibilities. > > > So let's settle this once and for all. Assuming that ZF is > > consistent, I ask: > > > 1. Is there a proof in ZF-Inf+~Inf that every set is finite? > > 2. Does there exist a model M of ZF-Inf+~Inf in which "there > > is an infinite set" is true? > > > Notice that exactly one of these questions has a "yes" answer > > and exactly one has a "no" answer. (Actually, come to think > > of it, since the base theory is ZF and not ZFC, it's possible > > that the answer to 1. is "yes" if by "finite" we mean one > > type of finite, say Dedekind finite, and "no" if we mean some > > other type of finite. In this case, I'd like to know which > > types of finite produce a "yes" answer.) > > > If 1. is "yes," then I hope that I will never again see a post > > claiming that ZF-Inf+~Inf doesn't prove that every set is in > > fact finite. In fact, I'll go as far as to suggest that if 1. > > is "yes," then those who claim that ZF-Inf+~Inf doesn't prove > > that every set is finite deserve to be called five-letter > > insults -- if posters are going to call those who deny the > > proof of Cantor's Theorem by five-letter insults, then those > > who deny the proof of "every set is finite" in ZF-Inf+~Inf > > also ought to be called the same. > > > If 2. is "yes," then what I'd like to know is how can I take > > _advantage_ of this fact? Suppose I want to consider a theory, > > based on ZF-Inf, which actually proves that an infinite set > > exists, yet also proves that no successor-inductive set > > containing 0 exists. > > > In the current Tony Orlow thread, there is a discussion about > > whether TO is defining N+ to be a successor-inductive set. It > > is possible that the theory that I mentioned above might be > > useful to discussing TO's ideas. > > > But of course, we can't proceed until we know, once and for > > all, whether ZF-Inf+~Inf proves every set to be finite or not. > > I posted a couple of articles in an old thread related to the > questions above, I will give references below. > > One issue in all this is how to define infinite in the posing of > the question. > > In usual ZF or ZFC, in the presence of the usual axiom of > infinity, we can prove there exists a smallest set having as > member emptyset and closed under the successor operation > on von Neumann ordinals. > > (The axiom of infinity does not directly assert the existence > of such a set. It asserts directly that there is a set having > emptyset as memeber and closed under trhe von Neumann successor > operation. It does not directly assert there is a smallest such > set. But that axiom of infinity together with the separation > axiom proves as a theorem there is a smallest such set.) > > We then define this smallest such set to be the set omega. > > The usual definition of finiteness in ordinary ZF or ZFC > is having von Neumann cardinaity a member of omega. Infinite > is defined as not finite in that sense. > > These definitions rest on the axiom of infinity itself, so > it at least bears some discussion whether to try working with > this definition or some alternatives. > > One alternative pair of definitions would be Dedekind finite > and Dedekind infinite. See my references below for more details. > > In my references below I also give another definition of finite, > not presupposing the usual axiom of infinity. > > With these definitions, we can get various rephrasings of the > question, that could be more suitable for this theory. > > My articles especially concentrate on Dedekind finite. But > from the discussion there you can also extract information > about the other cases. > > The articles: > > [1] David Libert "Axiom of Infinity (AxF) in ZF" > sci.logic Aug 26, 2003 > http://groups.google.com/group/sci.logic/msg/8b55a452fdf641d9 > > [2] David Libert "Axiom of Infinity (AxF) in ZF" > sci.logic Aug 29, 2003 > http://groups.google.com/group/sci.logic/msg/9417324657c5b242 Where is ZFC used to prove that if a system has a model then it is consistent? C-B > -- > David Libert ah...(a)FreeNet.Carleton.CA- Hide quoted text - > > - Show quoted text -
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 - |