details re Z-R |- PA consistent
in [Math]
|
From: MoeBlee on 6 Jul 2010 12:26 On Jul 5, 5:48 pm, RussellE <reaste...(a)gmail.com> wrote: > On Jul 5, 10:47 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > We define 'model of a theory'. Very roughly, a model of a theory T is > > a model FOR the language of T such that every member of T is mapped to > > the value 1 (1 for true, 0 for false) by a function that maps all > > sentences of the language of T to either 0 or 1 and as stipulated by > > the ordinary "Tarski method". > > Doesn't this mean the model is complete? I'm going to stop you right there. What do you mean by "model is complete"? Do you very clearly understand what a model is as I have defined it in my post and as my definition is found in Enderton's book? > "All sentences" means any wff? No, all sentences means all wffs that have no free variables. And in this context I'm referring to all sentences in the language of PA. I'll leave off the rest of your post for now. As I know a great amount of your posting history, I can tell you that you and I will not have a productive conversation if you don't familiarize yourself with a basic textbook on this subject (preferably Enderton, only because I used his particular formulations for some of the basic terminology). MoeBlee
From: Transfer Principle on 6 Jul 2010 15:55 On Jul 3, 2:05 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > Note: I hope to afford time to discuss this post with anyone who has > informed, coherent, and rational comments. But I might not defend this > post from uninformed, incoherent, and irrational criticisms, > especially from posters who have shown themselves over a good period > of time to be uneducable and hopeless cranks. Experience has shown > that certain of these people will not allow themselves to be properly > informed on certain matters in set theory and mathematical logic. That's funny. In this paragraph, MoeBlee tells us with which type of poster he wanted to have a discussion. And so rather than continue the discussion in the existing threads, he started this thread in the hopes that that other type of poster wouldn't notice this thread at all. But MoeBlee's plan failed. Nam Nguyen has noticed this thread, Russell Easterly has noticed this thread, and now I have noticed this thread. Three unwelcome posters whom MoeBlee was hoping wouldn't discover this thread now participate in this thread. Nice try, though, MoeBlee. In this "secret" thread which he tried in vain to keep hidden, MoeBlee provides us with a proof of Con(PA) in the theory "Z-R," or Z-Regularity. Sure enough, there isn't an "epsilon_0" to be found anywhere in the proof, hearkening back to the other thread in which I insisted that induction up to that ordinal was used in the proof. (In my defense, I wasn't the only poster who had mentioned epsilon_0 in that thread.) Once again, I don't deny that Con(PA) is provable in ZFC, or even Z-R, since Z-R proves the existence of omega (which MoeBlee writes as "w"), which is the (domain of the) model of PA that he constructs in the proof. But my comments about epsilon_0 and all that refer back to the mathematician Ed Nelson, who is working on a proof that PA is in fact inconsistent. I know that I've mentioned that proof attempt many times on sci.math, but now's as good a time as any to bring the link to the proof up yet again: http://www.math.princeton.edu/~nelson/papers/hm.pdf Nelson writes: "The goal is to produce an explicit superexponentially long recursion and prove that it does not terminate, thereby disproving Churchs Thesis from below, demonstrating that finitism is untenable, and proving that Peano Arithmetic is inconsistent." "Superexponentiation"? Also known as "tetration," Nelson defines this to be iterated exponentiation, in the same way that exponentiation is iterated multiplication. But epsilon_0 is defined to be the least infinite ordinal not obtainable from w via finite additions, multiplications, and exponentiations -- so superexponentiation is left out. To me, this leaves the door open for a proof of ~Con(PA) involving superexponentiation. A few other interesting Nelson quotes: "There is no clear concept of the finite in terms of which the infinite can be defined as not-finite." (Nelson, "Completed vs. Incomplete Infinity in Arithmetic") Hmmm. Doesn't that sound familiar? (Think AP.) "Now I live in a world in which there are no numbers save those that human beings on occasion construct." (Nelson, "Mathematics and Faith") Hmmm. Doesn't that sound familiar? (Think WM.) And of course, Srinivasan often refers to the theory IST, a theory created by -- you guessed it -- Ed Nelson. MoeBlee, Aatu, and the others can say however many times they want to about how PA is obviously consistent, and yet that hasn't stopped Nelson from searching for a proof that PA is instead inconsistent. And of course, since Z-R apparently proves that PA is consistent, if Nelson can prove that PA is inconsistent, then Z-R would be inconsistent as well. In conclusion, Nelson's work has the potential to vindicate those whom MoeBlee calls "hopeless cranks" across the board.
From: MoeBlee on 6 Jul 2010 16:03 On Jul 6, 2:55 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jul 3, 2:05 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > Note: I hope to afford time to discuss this post with anyone who has > > informed, coherent, and rational comments. But I might not defend this > > post from uninformed, incoherent, and irrational criticisms, > > especially from posters who have shown themselves over a good period > > of time to be uneducable and hopeless cranks. Experience has shown > > that certain of these people will not allow themselves to be properly > > informed on certain matters in set theory and mathematical logic. > > That's funny. In this paragraph, MoeBlee tells us with which type > of poster he wanted to have a discussion. And so rather than > continue the discussion in the existing threads, he started this > thread in the hopes that that other type of poster wouldn't > notice this thread at all. You're LYING. I did not hope that the thread would not be noticed by anyone at all. That would be a ridiculous hope. I've asked you many, many times, please stop LYING about me. > But MoeBlee's plan failed. Nam Nguyen has noticed this thread, > Russell Easterly has noticed this thread, and now I have noticed > this thread. Three unwelcome posters whom MoeBlee was hoping > wouldn't discover this thread now participate in this thread. You're CONTINUING your LIE. > In this "secret" thread which he tried in vain to keep hidden, > MoeBlee, Aatu, and the others can say however many times they > want to about how PA is obviously consistent, What SPECIFIC comment of mine do you have in mind where I mentioned 'obviousness'? > In conclusion, Nelson's work has the potential to vindicate > those whom MoeBlee calls "hopeless cranks" across the board. WHAT? I've told you before what I mean by 'crank'. It's a posting BEHAVIOR. It's not a matter of some OTHER mathematician devising this or that formal proof. And stop LYING about me. MoeBlee
From: herbzet on 6 Jul 2010 16:08 Transfer Principle wrote: > MoeBlee wrote: > > Note: I hope to afford time to discuss this post with anyone who has > > informed, coherent, and rational comments. But I might not defend this > > post from uninformed, incoherent, and irrational criticisms, > > especially from posters who have shown themselves over a good period > > of time to be uneducable and hopeless cranks. Experience has shown > > that certain of these people will not allow themselves to be properly > > informed on certain matters in set theory and mathematical logic. > > That's funny. In this paragraph, MoeBlee tells us with which type > of poster he wanted to have a discussion. And so rather than > continue the discussion in the existing threads, he started this > thread So far, so good. > in the hopes that that other type of poster wouldn't > notice this thread at all. Well, that's a ridiculous motive to impute to MoeBlee -- I hope he won't bother with refuting such a patent absurdity. -- hz
From: Transfer Principle on 6 Jul 2010 16:15
On Jul 5, 8:34 pm, William Hale <h...(a)tulane.edu> wrote: > I understand that you [Aatu] don't like the term "work in ZFC" since > mathematicians don't work in ZFC as such. But, I am using the term > "work" as a shorthand for what is meant when we say that ZFC serves as a > foundation for proving things in standard mathematics. > By standard mathematics, I was limiting myself to the areas of real > analysis, complex analysis, algebra, topology, and (differential) > geometry. That is, areas of mathematics discussed before 1900. I didn't > include logic since its main results were done after 1900. > Let me give some more on what I mean by "work in ZFC". I think some > non-mathematicians think that a mathematician comes up with an informal > proof and that it may or may not be formalized. I myself use the phrase "work in ZFC" all the time. But what do I mean by "work in ZFC"? Well, I consider those who _don't_ "work in ZFC" to be those who contradict it. Posters who claim that only countably many reals exist, such as Herc, don't "work in ZFC." Posters who claim no infinite sets exist, such as Srinivasan, don't "work in ZFC." Posters who claim that only finitely many naturals exist, such as WM, don't "work in ZFC." Posters who propose alternate theories and give such theories names, such as AP, TO, and tommy1729, don't "work in ZFC." By contrast, those who use ZFC to prove the above posters wrong are those who _do_ "work in ZFC." In a way, my definition of "work in ZFC" somewhat agrees with Hale's definition. Working with LCM's (in N) constitutes "working in PA" since PA proves the existence of LCM's, but working with LCM's in general rings where AC (Zorn's lemma) is needed constitutes "working in ZFC" instead. Doing classical analysis constitutes "working in ZFC" since ZFC formalizes classical analysis. On the other hand, working in a non-classical analysis where there are only countably many reals isn't "working in ZFC," since ZFC proves that there are uncountably many reals. |