God Proven by Known Laws of Physics and Theory of Everything

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Originally Posted by trish

The applicability of Godel’s theorem to TOE is an interesting one Brenda.

Godel’s theorem (the German spelling uses an umlaut over the o) says that that any recursively axiomatizable, consistent extension of Peano arithmetic is incomplete. What does this have to do with a Theory of Everything (TOE)?

Well since any lunacy can be deduced from an inconsistency, an inconsistent TOE would be useless. So if we construct a TOE, it will be consistent.

Also the whole idea of “having” a theory in hand means you can list the axioms. So any TOE we actually build will be recursively axiomizable.

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Gödel's theorem applies to this question when it is stated in this form... This is the one a professor told me to use to think about this problem.

If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.

In your reply you were commenting on "PA" and it's applicability. I agree PA has none however as I stated above the more general form of this theorem does. It applies to any and all mathematical theories.

The idea that Gödel's theorem applies to physics is not my own. See here.... This person seems to think it has a whole lot to do with the question. I think he may know what he is talking about this time.

thanks for the link, BrendaQG.

Your professor's statement (if read charitably) is essentially the same as the one i gave above.

Most non-mathematicians just assume an axiom system is presented recursively, so even though she/he left that out of the statement we can assume that she/he meant to include it. The sorts of theories of your professor is asking you consider are those which are capable of "talking about themselves." This means there are sentences in the language of the theory which can express things like "the theory is consistent", or "this very sentence is provable within the theory". [PA is a theory like this and so that's the point in having PA embeddable in your theory]. Godel showed that a recursively axiomatizable theory, capable of soundly expressing ideas about itself cannot both be consistent and complete. In fact, in a second theorem, Godel showed that if such a theory is consistent, then it won't be able to prove nor disprove the sentence which claims, "this theory is consistent". In short, a recursively presented, consistent theory of sufficiently expressivity will not admit a proof of its own consistency. Since there's a true proposition which it can't prove (namely its own consistency) such a theory is incomplete.

happy thinking.

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Originally Posted by trish

Hi Jamie.
I still have to differ with you on

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General relativity is not known to be inconsistent with quantum mechanics.

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Einstein’s classical theory of gravitation predicts a binary star system will lose energy at a continuous rate in the form of gravitational radiation. The prediction is at odds with quantum theory which holds that all mechanical systems gain or lose energy discretely. The two conceptions of nature are mutually incompatible. One of the theories must be dropped or modified. The procedures for modifying and adapting a classical field theory to quantum theory are somewhat standard, but far from universal; i.e. the fittings have to be handcrafted. For example, the correct Hamiltonian operator for the quantized field theory is usually inspired by but not determined uniquely by the form of the Hamiltonian for the classical field theory. So there are many different reasonable ways to attempt a quantization of the general relativity. The problem is none of the quantized theories of GR that have been constructed so far is consistent with nature. For example, the Feynman-Weinberg quantum gravity is not normalizable. ...

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Again, it's a popular misconception that general relativity contradicts quantum mechanics, since that's how it's often incorrectly described. But as Prof. John Donoghue and Dr. Tibor Torma have shown, the Feynman-Weinberg Lagrangian has not contradicted experiment, and it will not contradict experiment so long as the renormalized values of the infinite number of new coupling constants are adequately small. Additionally, as Steven Weinberg has stressed, this Lagrangian forms a theory of quantum gravity that is just as renormalizable as quantum electrodynamics and the Standard Model. For that, see:

John F. Donoghue, "General relativity as an effective field theory: The leading quantum corrections," Physical Review D, Vol. 50, Issue 6 (September 1994), pp. 3874-3888. Also at arXiv:gr-qc/9405057, May 25, 1994. http://arxiv.org/abs/gr-qc/9405057 (Note: download the PostScript version, since the PDF is corrupt.)

John F. Donoghue and Tibor Torma, "Power counting of loop diagrams in general relativity," Physical Review D, Vol. 54, Issue 8 (October 1996), pp. 4963-4972. Also released as "On the power counting of loop diagrams in general relativity," arXiv:hep-th/9602121, February 22, 1996. http://arxiv.org/abs/hep-th/9602121

Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations (Cambridge: Cambridge University Press, 1995), ISBN 0521550017, pp. 499 and 518-519.

For much more on this, I cordially invite you to read Prof. Frank J. Tipler's below paper:

F. J. Tipler, "The structure of the world from pure numbers," Reports on Progress in Physics, Vol. 68, No. 4 (April 2005), pp. 897-964. http://math.tulane.edu/~tipler/theoryofeverything.pdf Also released as "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything," arXiv:0704.3276, April 24, 2007. http://arxiv.org/abs/0704.3276

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... You have characterized the predicament this way:

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Standard quantum gravity (i.e., the Feynman-Weinberg Lagrangian) is known to be consistent with every experiment conducted to date and is (so far) internally consistent. But many physicists don't like it because it cannot be (completely) written down on paper, i.e., the series of individually-finite terms diverge to infinity. They're looking for something they can completely write down in a book, but they're looking in vain.

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It’s not really the case that mathematicians can’t write down a sum with infinitely many terms in a finite book. It’s easy enough to compactify notation with a summation sign or an integral sign. The problem is the sum diverges. ...

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I already know that. Below is the graphic that I made for the PARI/GP article on Wikipedia on December 5, 2006, which shows two different summations for producing two different irrational numbers (pi, and the base of the natural logarithm, e):

http://en.wikipedia.org/wiki/Image:P...Windows-XP.png

Of course, the exact number of iterations for producing those exact irrational numbers is infinity, but for the accuracy of the displayed significand the number of iterations chosen is enough to be accurate for every decimal place.

I also like creating (in the sense of formatting them in their respective system's language) algorithms for producing irrational numbers on my computer algebra systems (even though most of these systems have built-in fuctions for these), many of which use summations, definite integrals, product series, etc., with an infinite number of iterations. Below are some that I've created well before our discussion, where the proper number of iterations diverge to infinity (note that PARI/GP has no conception of "infinity," so simply replace that word with some arbitrarily high number):

Integral (Definite Integral)

PARI/GP:

sqrt(Pi) = intnum(x = -infinity, infinity, exp(-x^2))

Euler = -intnum(x = 0, infinity, exp(-x)*(log(x)))

Maxima:

sqrt(%pi) = bfloat(integrate(exp(-x^2), x, -inf, inf));

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Summation

PARI/GP:

exp(1) = sum(n=0, infinity, 1/n!)

Pi = sum(k=0, infinity, 2*(-1)^k*3^(1/2-k)/(2*k+1))

log(2) = sum(k=1, infinity, (-1)^(k-1)/k)

Euler = sum(k=1, infinity, (1/k)-log(1+1/k))

Maxima:

simpsum:true;

%e = sum(1/i!, i, 0, inf) = ~ bfloat(sum(1/i!, i, 0, 1024));

%pi^2/6 = bfloat(sum(1/(n^2), n, 1, inf));

%pi = sum(2*(-1)^k*3^(1/2-k)/(2*k+1), k, 0, inf); = ~ bfloat(sum(2*(-1)^k*3^(1/2-k)/(2*k+1), k, 0, 4096));

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Product

PARI/GP:

1/sqrt(2) = prod(x = 0, infinity, 1-(1/(4*x+2)^2))+0.

Maxima:

1/sqrt(2) = product(1-(1/(4*i+2)^2), i, 0, inf);

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Originally Posted by Jamie Michelle

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Originally Posted by trish

Hi Jamie.
I still have to differ with you on

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General relativity is not known to be inconsistent with quantum mechanics.

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Einstein’s classical theory of gravitation predicts a binary star system will lose energy at a continuous rate in the form of gravitational radiation. The prediction is at odds with quantum theory which holds that all mechanical systems gain or lose energy discretely. The two conceptions of nature are mutually incompatible. One of the theories must be dropped or modified. The procedures for modifying and adapting a classical field theory to quantum theory are somewhat standard, but far from universal; i.e. the fittings have to be handcrafted. For example, the correct Hamiltonian operator for the quantized field theory is usually inspired by but not determined uniquely by the form of the Hamiltonian for the classical field theory. So there are many different reasonable ways to attempt a quantization of the general relativity. The problem is none of the quantized theories of GR that have been constructed so far is consistent with nature. For example, the Feynman-Weinberg quantum gravity is not normalizable. ...

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Again, it's a popular misconception that general relativity contradicts quantum mechanics, since that's how it's often incorrectly described. But as Prof. John Donoghue and Dr. Tibor Torma have shown, the Feynman-Weinberg Lagrangian has not contradicted experiment, and it will not contradict experiment so long as the renormalized values of the infinite number of new coupling constants are adequately small. ...

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And how does this address the issue of continuous orbital decay as opposed to discrete orbital decay? It doesn't. We don't need experimental evidence that two theories are mutually incompatible when they make conflicting predictions. When (or if) we have experimental evidence of the existence of gravitons, then we'll have experimental evidence of the conflict between GR and QFT, if you so require it.

That is not to say some modification of GR won't be found that's compatible with QFT...that is everybody's hope afterall. It hasn't been found yet. You said yourself that in Tipler's theory the relevant series diverge. Let's use the example of a satellite again. If the series expansion for the wave function of a satellite diverges, then the theory can make no predictions about the position of the satellite. You can add up the first 100 terms and base a prediction on that, but that would your prediction, not the prediction of the theory (which requires all the terms to be summed). If you add up the first thousand terms instead, you'll get a different prediction because the series is not stabilizing, it is...afterall...diverging.

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Originally Posted by trish

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Originally Posted by Jamie Michelle

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Originally Posted by trish

Hi Jamie.
I still have to differ with you on

Quote:

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General relativity is not known to be inconsistent with quantum mechanics.

---

Einstein’s classical theory of gravitation predicts a binary star system will lose energy at a continuous rate in the form of gravitational radiation. The prediction is at odds with quantum theory which holds that all mechanical systems gain or lose energy discretely. The two conceptions of nature are mutually incompatible. One of the theories must be dropped or modified. The procedures for modifying and adapting a classical field theory to quantum theory are somewhat standard, but far from universal; i.e. the fittings have to be handcrafted. For example, the correct Hamiltonian operator for the quantized field theory is usually inspired by but not determined uniquely by the form of the Hamiltonian for the classical field theory. So there are many different reasonable ways to attempt a quantization of the general relativity. The problem is none of the quantized theories of GR that have been constructed so far is consistent with nature. For example, the Feynman-Weinberg quantum gravity is not normalizable. ...

---

Again, it's a popular misconception that general relativity contradicts quantum mechanics, since that's how it's often incorrectly described. But as Prof. John Donoghue and Dr. Tibor Torma have shown, the Feynman-Weinberg Lagrangian has not contradicted experiment, and it will not contradict experiment so long as the renormalized values of the infinite number of new coupling constants are adequately small. ...

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And how does this address the issue of continuous orbital decay as opposed to discrete orbital decay? It doesn't. We don't need experimental evidence that two theories are mutually incompatible when they make conflicting predictions. When (or if) we have experimental evidence of the existence of gravitons, then we'll have experimental evidence of the conflict between GR and QFT, if you so require it.

That is not to say some modification of GR won't be found that's compatible with QFT...that is everybody's hope afterall. It hasn't been found yet. You said yourself that in Tipler's theory the relevant series diverge. Let's use the example of a satellite again. If the series expansion for the wave function of a satellite diverges, then the theory can make no predictions about the position of the satellite. You can add up the first 100 terms and base a prediction on that, but that would your prediction, not the prediction of the theory (which requires all the terms to be summed). If you add up the first thousand terms instead, you'll get a different prediction because the series is not stabilizing, it is...afterall...diverging.

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It addresses it because that's incorrect. That's an incorrect way that these GR/QM issues are often put, i.e., saying that they contradict each other when really it's that the maths diverge to arbitrarily high orders of derivatives in order to be consistent. As Prof. John Donoghue and Dr. Tibor Torma have shown, the Feynman-Weinberg Lagrangian has not contradicted experiment, and it will not contradict experiment so long as the renormalized values of the infinite number of new coupling constants are adequately small. Additionally, as Steven Weinberg has stressed, this Lagrangian forms a theory of quantum gravity that is just as renormalizable as quantum electrodynamics and the Standard Model. For that, see:

John F. Donoghue, "General relativity as an effective field theory: The leading quantum corrections," Physical Review D, Vol. 50, Issue 6 (September 1994), pp. 3874-3888. Also at arXiv:gr-qc/9405057, May 25, 1994. http://arxiv.org/abs/gr-qc/9405057 (Note: download the PostScript version, since the PDF is corrupt.)

John F. Donoghue and Tibor Torma, "Power counting of loop diagrams in general relativity," Physical Review D, Vol. 54, Issue 8 (October 1996), pp. 4963-4972. Also released as "On the power counting of loop diagrams in general relativity," arXiv:hep-th/9602121, February 22, 1996. http://arxiv.org/abs/hep-th/9602121

Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations (Cambridge: Cambridge University Press, 1995), ISBN 0521550017, pp. 499 and 518-519.

For much more on this, I cordially invite you to read Prof. Frank J. Tipler's below paper:

F. J. Tipler, "The structure of the world from pure numbers," Reports on Progress in Physics, Vol. 68, No. 4 (April 2005), pp. 897-964. http://math.tulane.edu/~tipler/theoryofeverything.pdf Also released as "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything," arXiv:0704.3276, April 24, 2007. http://arxiv.org/abs/0704.3276

Jamie, it's not incorrect that GR predicts a continuous orbital decay. it's not incorrect that QFT requires a discrete orbital decay.

what could it possibly mean to say

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...saying that they contradict each other when really it's that the maths diverge to arbitrarily high orders of derivatives in order to be consistent.

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do you mean that if one insists on too much accuracy the two theories will make different predictions?

or do you mean that if one insists on a quantum theory of gravity (QTOG) that is actually renormalizable, then it won't be consistent.

in the former case all i can say is: well yeah, that's because the two theories are inconsistent.

in the latter case, you've got a theory that useless because it won't make predictions. you also, by the way don't have GR anymore. you have a different theory. a modified GR. a QTOG, if you will. you keep insisting GR+QTF is consistent, but you keep showing us QTOG which is itself inconsistent with GR.

I think I need some crackers to go along with this mathematical alphabet soup.

something wicked this way comes. http://news.bbc.co.uk/2/hi/science/nature/7184521.stm

Yes, hippi....I'm lost too. Looks like I'll have to sign up for an advanced physics course to keep up with Jamie and Trish. Renormalization is a kind of mathematical trickery to prove something that can't be proven any other way.

HERE'S A SHORT SYNOPSIS OF MY SLANTED VIEW OF THE LAST COUPLE OF PAGES FOR THE REST OF YOU WHO "CLAIM" NOT TO FOLLOW:

Renormalization in a Nutshell:
Quantum theory never makes predictions like "the proton will have location X at time T; instead it allows one to compute the probability P that the proton will be at X when the clock says T. A probability by definition is a number between zero and one. Unfortunately, sometimes in an attempt to construct a quantum theory to model a particular situation a theoretician will get an embarrassing answer like P(E) = 8974.553. When this happens we say his model needs to be normalized. There has to be a uniform way to fix all these erroneous calculations at once to get answers in the correct range from zero to one. Sometimes it's just as simple as dividing everything by just the right number. If everything can be fixed we say the theory is renormalizable. Sometimes it can can be shown there's no way to fix the model. This is the case whenever you're getting results like P = infinity. In these cases we say the model is not renormalizable. When that happens the theoretician has to go back to the drawing board and try to reformulate, revise and redesign her approach.

The Situation with Quantum Gravity:
Nobody has yet found a renormalizable theory of quantum gravity. It's not clear to me, after all this discussion, whether Jamie claims otherwise, or is claiming renormalization doesn't matter.

The Need for a Quantum Theory of Gravity:
Everyone thinks we need one because the best theory of gravity we have to date (which is in my opinion every bit as beautiful as Allanah dressed as cleopatra)
makes predictions about the nature of gravitational radiation which are at odds with quantum theory. In particular quantum theory claims that the energy carried gravitational wave of a fixed frequency can only take on discrete values. General relativity says all values are possible. One consequence of this is general relativity predicts nothing can escape a black hole. Quantum theory predict black holes eventually evaporate and everything escapes. The general feeling is: these two theories need to be reconciled.