How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent(EASY PROOF)
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From: Nam Nguyen on 8 Jul 2010 21:50 Transfer Principle wrote: > On Jul 7, 8:58 pm, Marshall <marshall.spi...(a)gmail.com> wrote: >> On Jul 7, 7:22 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>> That's all the _technical_ arguments an intellectual clown like you, Marshall, >>> could ever say! >> PA is provably consistent. Learn why, or shut [...] up. > > Yet that hasn't stopped the mathematician Ed Nelson from > searching for a proof that PA is inconsistent. > >> Give it up, loon. PA is provably consistent. Deal with it. > > If those who even entertain the possibility that PA is > inconsistent are "loons," then I guess that makes Nelson > and Nguyen (and myself, since I keep bringing up Nelson) > a bunch of "loons." For the record, I've never suggested that PA is _actually_ inconsistent. (It's just that neither have I suggested it's _actually_ consistent). To me, asserting it's actually one way or the other as if we were privy to such knowledge would be extreme positions. Perhaps I could be left out of the arguments between these 2 sides? > > It would be poetic justice for Nelson to complete his proof > and someone to tell Spight: > > "PA is provably _in_consistent. Learn why, or shut up!" -- ---------------------------------------------------- There is no remainder in the mathematics of infinity. NYOGEN SENZAKI ----------------------------------------------------
From: Nam Nguyen on 9 Jul 2010 23:47 Nam Nguyen wrote: > David Libert wrote: >>[...] > we trust our intuition but if necessary we'll verify that with > the rigorousness [of] proofs through rules of inference. If for > whatever the reason we couldn't [do] the verification then the > trusted is always an un-verified hence a guarded trust only. > > *** > > And that's just my high-level of the situation. In details I do have > a few technical reasons that would reveal unsettling reasons why the > compatibility between intuition and rigor of syntactical proof is > very much questionable. The 1st incompatibility is that while formal provability is relative to axiom formal systems, intuition truths are purported to be non- relativistic and absolute: a formula is "true" because its meaning is supposed to be _always true_ which wouldn't change just because we might look at different axiom formal systems! This would make intuition knowledge unfit as a mechanism of judging matters of syntactical provability, such as say syntactical consistency in general or PA's consistency in particular, simply because truth and falsehood are _intrinsically independent_ from provability. That's why a formula's truth or falsehood is always a _mental, meta, subjective mapping_ rather than a resultant of a _mechanical, systematical_ process such as provability through rules of inference. *** The 2nd incompatibility is that while intuition about a formula truth is "binary", in the sense we'd intuit the meaning of a formula as true or false and if it's true then it's impossible for us to imagine it as false - _or any other value_ - and vice versa, intuitions about provability (even in one axiom system) are not necessarily "binary" in general, in the sense if we intuit that there's no proof for a formula, there could be actually 2 _distinct valid reasons_ why there's no proof: the "proof" length is zero, or is infinite! The consequence of this incompatibility is that if we happen to investigate in a system T a formula F which falls into this category in which it's genuinely impossible to know if it's the case of the 0-"proof"-length or the infinite-"proof"-length then any intuition about the models of this T or about the overall knowledge of formula truths about this T must necessarily be incomplete. Consequently, it'd, together with the 1st incompatibility, reinforce the notion we can't really know provability matter such as T's (in)consistency through our intuitive knowledge of truths. An example of such F and such T are pGC and PA. And the knowledge of the natural numbers is then too incompatible with the syntactical proof mechanism to really be capable of asserting the consistency of PA (if PA is actually consistent in the first place). *** Put it differently, these 2 incompatibilities means that the intuitive concept of "the natural numbers" could equally be any one of the following concepts(and certainly there would be more): - one in which cCG is true; - one in which ~cGC is true; - one in which G(PA) is true; - one in which ~(PA) is true; - one in which cGC <-> G(PA) is true - one in which ~cGC <-> G(PA) is true - infinitely many more concepts ... This is in essence what the proposed Principle of Symmetry is about. And we're certain no more being a "crank" than being a "standard theorist" if we pick any of these, e.g. "one in which ~(PA) is true", to be "the natural numbers". It's all a matter of relative, subjective choice of concept, and not a matter of (an absolute) knowledge, as those who have argued against my position tend to believe (Aatu, MoeBlee, ... iirc). -- ---------------------------------------------------- There is no remainder in the mathematics of infinity. NYOGEN SENZAKI ----------------------------------------------------
From: Nam Nguyen on 9 Jul 2010 23:51 Nam Nguyen wrote: > Nam Nguyen wrote: >> David Libert wrote: >>> [...] > > we trust our intuition but if necessary we'll verify that with > > the rigorousness [of] proofs through rules of inference. If for > > whatever the reason we couldn't [do] the verification then the > > trusted is always an un-verified hence a guarded trust only. >> >> *** >> >> And that's just my high-level of the situation. In details I do have >> a few technical reasons that would reveal unsettling reasons why the >> compatibility between intuition and rigor of syntactical proof is >> very much questionable. > > The 1st incompatibility is that while formal provability is relative > to axiom formal systems, intuition truths are purported to be non- > relativistic and absolute: a formula is "true" because its meaning is > supposed to be _always true_ which wouldn't change just because we might > look at different axiom formal systems! > > This would make intuition knowledge unfit as a mechanism of judging matters > of syntactical provability, such as say syntactical consistency in general > or PA's consistency in particular, simply because truth and falsehood are > _intrinsically independent_ from provability. That's why a formula's truth > or falsehood is always a _mental, meta, subjective mapping_ rather than > a resultant of a _mechanical, systematical_ process such as provability > through rules of inference. > > *** > > The 2nd incompatibility is that while intuition about a formula truth > is "binary", in the sense we'd intuit the meaning of a formula as true > or false and if it's true then it's impossible for us to imagine it > as false - _or any other value_ - and vice versa, intuitions about > provability (even in one axiom system) are not necessarily "binary" > in general, in the sense if we intuit that there's no proof for a > formula, there could be actually 2 _distinct valid reasons_ why there's > no proof: the "proof" length is zero, or is infinite! > > The consequence of this incompatibility is that if we happen to > investigate in a system T a formula F which falls into this category > in which it's genuinely impossible to know if it's the case of the > 0-"proof"-length or the infinite-"proof"-length then any intuition > about the models of this T or about the overall knowledge of formula > truths about this T must necessarily be incomplete. Consequently, it'd, > together with the 1st incompatibility, reinforce the notion we can't > really know provability matter such as T's (in)consistency through our > intuitive knowledge of truths. > > An example of such F and such T are pGC and PA. And the knowledge of > the natural numbers is then too incompatible with the syntactical proof > mechanism to really be capable of asserting the consistency of PA (if > PA is actually consistent in the first place). > > *** > > Put it differently, these 2 incompatibilities means that the intuitive > concept of "the natural numbers" could equally be any one of the > following concepts(and certainly there would be more): > > - one in which cCG is true; > - one in which ~cGC is true; > - one in which G(PA) is true; > - one in which ~(PA) is true; I meant: "- one in which ~g(PA) is true;" > - one in which cGC <-> G(PA) is true > - one in which ~cGC <-> G(PA) is true > - infinitely many more concepts ... > > This is in essence what the proposed Principle of Symmetry is about. > And we're certain no more being a "crank" than being a "standard > theorist" if we pick any of these, e.g. "one in which ~(PA) is true", > to be "the natural numbers". > > It's all a matter of relative, subjective choice of concept, and not > a matter of (an absolute) knowledge, as those who have argued against > my position tend to believe (Aatu, MoeBlee, ... iirc). -- ---------------------------------------------------- There is no remainder in the mathematics of infinity. NYOGEN SENZAKI ----------------------------------------------------
From: Nam Nguyen on 10 Jul 2010 14:32
Nam Nguyen wrote: > MoeBlee wrote: >> >> I do not know that there is no consistent theory that proves PA is >> consistent. > > I take it that you meant to answer you don't know the answer to this > question. (But you have to let me know if this is the case). > > (If I'm to answer the question then my answer is NO: there can be no > such formal proof.) > > >> And with a theory that has a sole non-logical axiom that is a >> formalization of "PA is consistent", we do have a consistent theory >> that proves PA is consistent. But, AGAIN, I don't draw any >> epistemological import (basis for belief that PA is consistent) from >> such a thing. >> >> And aside from such theories as just mentioned, I do have basis to >> believe (without claiming certainty) that there exists a (non-trivial) >> consistent theory that proves PA is consistent. However, I do not take >> such a proof itself as a basis that one should believe PA is >> consistent if one already had strong doubts that PA is consistent. >> >> And I've explained about all of that in previous posts (either in this >> thread or in previous threads). Please read Franzen's book; as it >> would save a lot of typing for both of us. > > I did read the portion of the book I have and from that I already > concluded that TF actually didn't have an answer to the question > I asked you above. If you believe he had done so, and I'm wrong > here, you had to cite _a clear reference_ in his book where he'd > answer the question somehow. A mere mentioning "Franzen's book" > isn't sufficient to justify to an answering the question I asked. Since it didn't seem you were able to cite *a clear reference* that you had said "would save a lot of typing for both of us", I'll cite a reference to point out it wouldn't at all save us time to leverage what he wrote there. Torkel Franzen wrote in a book: > In a mathematical context, on the other hand, mathematicians easily > speak of truth. How is easy is "easily". That, MoeBlee, is *just one man's opinion* and *not a mathematical fact*, especially when no one has a slightest idea about, e.g., the truth value of cGC! > "If the generalized Riemann hypothesis is true...", > "There are strong grounds for believing that Goldbach�s conjecture is > true...", "If the twin prime conjecture is true, there are infinitely > many counterexamples...." In such contexts, the assumption that an > arithmetical statement is true is not an assumption about what can be > proved in any formal system, or about what can be "seen to be true", > and nor is it an assumption presupposing any dubious metaphysics. OK so far. But did you notice the key word _assumption_ ? > Rather, the assumption that Goldbach�s conjecture is true is exactly > equivalent to the assumption that every even number greater than 2 is > the sum of two primes, the assumption that the twin prime conjecture is > true means no more and no less than the assumption that there are > infinitely many primes p such that p+2 is also a prime, and so on. Again, the key word _assumption_ - _not fact_ ! > In other words "the twin prime conjecture is true" is simply another > way of saying exactly what the twin prime conjecture says. I'm sorry MoeBlee, such passages fails to make a _clear distinction_ between the purported "semantic" of a formula (which would be just one out of infinitely many possible semantics) and the purported "truth" of a formula. The negation "the twin prime conjecture is true" has a meaning (it's meaningful) too, just like 1+1=/=0, or 1+1=0, etc... mathematically speaking. Why suddenly, only "the twin prime conjecture is true" or "1+1=0 is true" e.g. are true? I don't think you could answer those questions on Torkel, right? > It is a mathematical statement, not a statement about what can be > known or proved, or about any relation between language and a > mathematical reality. Every mathematical statement has its negation. No? By what criteria that one of the 2 is true and the other isn't? By the criteria stipulated by only a few of us (Torkel, Aatu, MoeBlee, ...)? If so, wouldn't you agree "ordinary" mathematical reasoning and proofs (as oppose to syntactical ones through rules of inference) are just subjective and relative, which is what I've been saying all along? -- ---------------------------------------------------- There is no remainder in the mathematics of infinity. NYOGEN SENZAKI ---------------------------------------------------- |