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 17:14 On Jun 29, 4:05 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > What do you mean "obviously"? There are very few set theorists to > believe it the case. Sorry, delete that. I cut your sentence off and unintentionally misconstrued you. MoeBlee
From: Transfer Principle on 29 Jun 2010 17:23 On Jun 29, 9:30 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > On Jun 29, 12:13 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > If you still can't see it, then, if I'm feeling generous, I'll outline it > > for you. As to showing an exact sequence of primitive formulas of the > > language of Z, no, that's just a chore. > I don't know what you're referring to. I did ask for the statement of > the theorem in ZFC, but nobody has come up with that either. > So in summary, > 1. ZFC can prove PA consistent - it's easy and lots of people have > done it. > 2. Nobody can give a reference to its being done. > 3. Nobody can describe the proof that has been done in ZFC. > 4. Nobody can give even the ZFC expression for the theorem itself. > In other words, business as usual. I'm not skeptical (in the way that Charlie-Boo is skeptical) about the provability of Con(PA) in either ZFC or PRA+epsilon_0. But I am suspicious of the fact that the induction needs only to be taken up to epsilon_0, which the the smallest ordinal not reachable from omega via finitely many additions, multiplications, and exponentiations, but can be reached via finitely many _tetrations_ since epsilon_0 = omega^^omega. This is why I so often mention Ed Nelson and his proof attempt of ~Con(PA) involving tetration.
From: MoeBlee on 29 Jun 2010 17:28 On Jun 29, 4:23 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > I'm not skeptical (in the way that Charlie-Boo is skeptical) about > the provability of Con(PA) in either ZFC or PRA+epsilon_0. But I > am suspicious of the fact that the induction needs only to be > taken up to epsilon_0, As to ZFC, you don't need such fancy stuff as epsilon_0. Just do the routine proof that with the system of omega with 0, successor, addition, and multiplication we get a model of all the PA axioms. MoeBlee
From: Transfer Principle on 29 Jun 2010 21:17 On Jun 29, 2:05 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 29, 3:44 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > 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. In that case, if Srinivasan were to work in NBG-Infinity instead of ZF-Infinity, would he then be allowed to talk about his "D"? For I see no reason that D wouldn't be a class in NBG-Infinity (but then, whether this class is a _proper_ class or the empty set depends on the axiom that Srinivasan wishes to add to it).
From: herbzet on 29 Jun 2010 21:54
herbzet wrote: > ZFC is certainly powerful enough I think the phrase I wanted was "expressive enough". > to talk about itself as well > as to talk about PA, so it's not inherently contradictory for > ZFC to provide a formal proof of this or that about itself. |