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 29 Jun 2010 16:36 On Jun 29, 12:28 pm, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote to much flotsam still for me to spend more time than I've already spent. However: > > We PROVE from ZF-Inf that there IS NO SUCH object that you are calling > '> D'. (or at least we have not before us a proof that there IS such an > > object). Just adding a constant symbol 'D' and saying whategver you > > want about it does not override. > You do not have any such proof. I SAID, "or at least we have not before us a proof that there IS such an object". But it's simple anyway: Therorem of ZF-I: Ex~En x in P_n(0) -> ~EyAz(zey <-> ~En z in P_n(0)) Proof: Toward a contradiction suppose Ex~En x in P_n(0) and Az(zey <-> ~En x in P_n(0)). Let ~En x in P_n(0). Let j be arbitrary. ~En xu{j} in P_n(0). So Aj j in Uy. Theorem of ZF-I: ~Ex~En x in P_n(0) -> Ey(Az(zey <-> ~En z in P_n(0)) & y=0) Proof: Immediate. Then, as far as I know (which is pretty limited) it is not decided in ZF-I whether Ex~En x in P_n(0). Someone may inform me further on that, but I'm pretty sure that ZF-I doesn't tell us whether there are or are not sets other than the hereditarily finite sets. > How can something be true > "in the standard model of PA iff PA is inconsistent" ????? Typo of omission. I meant, "true in the standard model for the LANGUAGE of PA", as I had posted in previous messages. MoeBlee
From: MoeBlee on 29 Jun 2010 16:43 On Jun 29, 12:28 pm, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote: > On Jun 29, 8:33 pm, MoeBlee <jazzm...(a)hotmail.com> wrote:> On Jun 29, > > > I do not want to allow any explicit references to infinite objects in > > > the language of the theory ZF-Inf. > > > What does that MEAN? What is your technical definition of "explict > > reference to infinite sets in the language"? > > It is a phrase in good old English. I am saying that no definitions of > any infinite objects are allowed in the language of ZF-Inf and no > symbol in the language of ZF-Inf represents any infinite set. For > example "the set of PA-sentences" cannot even be defined in the > language of ZF-Inf under this constraint. This is the best I can do > with explanations. That such a language would be legitimate is obvious > to me. That's ignorance and confusion (some of it CORRECT, but APPLIED incorrectly). Each day, like a new batch of tarballs on a Gulf coast beach. Just not enough time in the day to clean it all up. MoeBlee > > > That theory entails that every object is finite. And there is no > > definition of any infinite object possible in that theory. >
From: Transfer Principle on 29 Jun 2010 16:44 On Jun 29, 10:28 am, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote: > On Jun 29, 8:33 pm, MoeBlee <jazzm...(a)hotmail.com> wrote:> On Jun 29, 2:09 am, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote: > > We PROVE from ZF-Inf that there IS NO SUCH object that you are calling > > 'D'. (or at least we have not before us a proof that there IS such an > > object). Just adding a constant symbol 'D' and saying whategver you > > want about it does not override. > You do not have any such proof. You don't even begin to understand > what I am talking about here. But I have attempted to give an > explanation further down. This big argument between MoeBlee and Srinivasan over what exactly this "D" is reminds me of the mathematician Goedel and his theory ZF+"V=L." So if MoeBlee is going to ask Srinivisan about "D," then maybe we should be asking about Goedel's "V" and "L." So what exactly are "V" and "L" anyway? V is supposed to be the universe of all sets. So V is too large to be a set. Thus, if MoeBlee is going to criticize Srinivasan for not proving that D is a set, maybe he should look at Goedel's V. What about "L"? L is supposed to be the constructible universe, and obviously if V=L and V is too large to be a set, then L must be too large to be a set. Even if ~V=L, I've heard that L contains all ordinals, and the ordinals are too large to be a set, and so L must be too large to be a set no matter what. On the contrary, if D=0 as asserted by Srinivasan, then D must be a set since 0 (the empty set) is evidently a set as well. Of course, we could call V and L "proper classes," but then again, MoeBlee has already pointed out that in ZF (unlike NBG) there are no proper classes. Thus, we can't talk about V and L, yet that didn't stop Goedel from writing the axiom "V=L." Therefore, Srinivasan's statement "D=0" is a valid axiom if and only if Goedel's statement "V=L" is a valid axiom. If MoeBlee is going to insist that Srinivasan prove that D is actually a set, then maybe MoeBlee should do the same for Goedel's V and L.
From: Frederick Williams on 29 Jun 2010 17:01 Transfer Principle wrote: > > Of course, we could call V and L "proper classes," but then > again, MoeBlee has already pointed out that in ZF (unlike NBG) > there are no proper classes. Thus, we can't talk about V and L, > yet that didn't stop Goedel from writing the axiom "V=L." Which set theory do you think G"odel worked with? Also, though it is true that there are no proper sets in ZF, there are formulae with one free variable. -- I can't go on, I'll go on.
From: MoeBlee on 29 Jun 2010 17:05
On Jun 29, 3:44 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 29, 10:28 am, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote: > > > On Jun 29, 8:33 pm, MoeBlee <jazzm...(a)hotmail.com> wrote:> On Jun 29, 2:09 am, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote: > > > We PROVE from ZF-Inf that there IS NO SUCH object that you are calling > > > 'D'. (or at least we have not before us a proof that there IS such an > > > object). Just adding a constant symbol 'D' and saying whategver you > > > want about it does not override. > > You do not have any such proof. You don't even begin to understand > > what I am talking about here. But I have attempted to give an > > explanation further down. > > This big argument between MoeBlee and Srinivasan over what > exactly this "D" is reminds me of the mathematician Goedel > and his theory ZF+"V=L." So if MoeBlee is going to ask > Srinivisan about "D," then maybe we should be asking about > Goedel's "V" and "L." Why? (1) Godel worked in NBG, in which there DO exist proper classes. (2) Even those who work in Z set theories often note that we mention proper classes such as V and L NOT as if the theory proves there are objects matching the definiens of 'V' and 'L' but rather as shortcut language in the meta-theory and that mention of such proper classes can be reduced to rubric without proper classes. > So what exactly are "V" and "L" anyway? V is supposed to be > the universe of all sets. So V is too large to be a set. Thus, if > MoeBlee is going to criticize Srinivasan for not proving that D > is a set, maybe he should look at Goedel's V. No, you don't know what you're talking about. IN Z-Inf, I just posted what happens with "D". Meanwhile, Godel was in NBG in which theory we DO prove the existence of certain proper classes. > What about "L"? L is supposed to be the constructible universe, > and obviously if V=L What do you mean "obviously"? There are very few set theorists to believe it the case. > and V is too large to be a set, then L must > be too large to be a set. So what? They're proper classes that are proven to exist in NBG. And in Z, the expression "V=L" is not one actually in the language of ZF but rather a nickname for an actual formula that is in the language of ZF. > Even if ~V=L, I've heard that L contains > all ordinals, and the ordinals are too large to be a set, and so L > must be too large to be a set no matter what. So what? It's a proper class, and (though I haven't personally worked through all the details), proven to exist in NBG. > On the contrary, > if D=0 as asserted by Srinivasan, then D must be a set since 0 > (the empty set) is evidently a set as well. I just posted a post that clarifies Srinivasan's defiens for 'D'. > Of course, we could call V and L "proper classes," but then > again, MoeBlee has already pointed out that in ZF (unlike NBG) > there are no proper classes. Thus, we can't talk about V and L, > yet that didn't stop Goedel from writing the axiom "V=L." Because Godel was in NBG!!! What the hell is your problem?!!! Also, even if in ZF, we refer to V and L as "figures of speech" that must resolve back to actual formulas in the language of ZF. > Therefore, Srinivasan's statement "D=0" is a valid axiom if and > only if Goedel's statement "V=L" is a valid axiom. No it is not. See my other post. > If MoeBlee is > going to insist that Srinivasan prove that D is actually a set, then > maybe MoeBlee should do the same for Goedel's V and L. You pontificate out of IGNORANCE. Godel worked in NBG where we prove that there do exist proper classes. Also, even if in ZF, we refer to V and L as "figures of speech" that must resolve back to actual formulas in the language of ZF. Christ man, would you get HOLD of yourself? MoeBlee |