How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: Charlie-Boo on 1 Jul 2010 00:14 On Jun 29, 5:23 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > 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. Do you have a reference that shows a proof that PA is consistent, carried out entirely in ZFC? If so, what is the formal expression in ZFC that PA is consistent? How many lines are there in the proof? Which ZFC axioms are used that PA would need to carry out the proof entirely in PA? C-B
From: Charlie-Boo on 1 Jul 2010 00:17 On Jun 29, 5:28 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > 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. The problem isn't to conclude that a model exists, using ZFC. The problem is to prove that PA is consistent, using ZFC. Another question: How many ways can people propose something other than a reference to a proof in ZFC that PA is consistent? 1. Reference doesn't have it. 2. No reference. 3. Talks about a proof of something else in ZFC. 4. Isn't carried out in ZFC. (not mutually exclusive)etc. C-B > MoeBlee
From: Charlie-Boo on 1 Jul 2010 00:23 On Jun 29, 9:51 pm, herbzet <herb...(a)gmail.com> wrote: > "Jesse F. Hughes" wrote: > > Of course, I'm really here for lower entertainment. I want posts about > > the Hammer, about how surrogate factoring moves the stock market, about > > the most influential mathematicians on the planet. But still I pretend > > to care about arguments, if only for appearance's sake. > > Well, I didn't come here for the low comedy -- that's simply not an idea > that had occurred to me. But, it being made explicitly an option -- I > suppose it's a valid choice, even one of some value. > > If I have any reservation, it would just be that I don't think that > encouraging crankery for its entertainment value crank: "an unbalanced person who is overzealous in the advocacy of a private cause" And what are signs of being unbalanced? How about unsubstantiated beliefs i.e. delusions? And what is overzealous? How about lying about what a reference contains and making personal attacks on people? And is any cause truly private - isn't it just a matter of degree, of how many people believe it? C-B > is such a great idea, > all things considered. Not that you're suggesting that. > > Nevertheless, thanks for enlarging the realm of possibilities. > > -- > hz
From: Charlie-Boo on 1 Jul 2010 00:34 On Jun 30, 5:23 am, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > Charlie-Boo wrote: > > The best way to explain that a ZFC axiom is not used is to give the > > proof without using any ZFC axioms - good luck! > > > How would you prove the PA axioms in ZFC, then? You keep saying it > > isn't from an axiom but can't say how it is done - so how do you know > > it isn't? > > As Tim Little says elsewhere in the thread, you define S, +, and * in > the language of ZFC and then prove the counterparts of PA's axioms as > theorems. It's done in Suppes (probably without C). Which requires the ZFC equivalent of Peano's Axioms (the axiom of infinity.) C-B > -- > I can't go on, I'll go on.
From: George Greene on 1 Jul 2010 01:12
On Jul 1, 12:34 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > Which requires the ZFC equivalent of Peano's Axioms (the axiom of > infinity.) You are not JUST wrong, you are the EXACT OPPOSITE of right. It's like the right answer to the question was 3, but instead of saying 1, or 2, or 0, or 5, or 42, or 69, you said NEGATIVE 3. |