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Impact of Gödel's incompleteness theorems on a TOE 
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#73
Dec1310, 06:55 PM

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#74
Dec1310, 07:06 PM

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#75
Dec1310, 07:06 PM

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the tape of any TM that halts is bound by the busy beaver function. It's large, it's unbounded, and it's finite. If the TM doesn't halt, you don't need any tape at all, because you'll never get the result of the TM which is define as what remains when the TM stops. Of course you can't, in general, know if your TM belong to one or the other class. Doesn't change the tape you need is finite. if you think it's infinite, then precise what infinite you're talking about. Is it aleph 0? No, or you'd enter hypercomputation. Is it more? No, that's worse! Is it less than aleph 0? Well, if you believe there is such a thing as an infinite less than aleph 0, go and publish! Sorry, I'm begining to be fed up of stating the obvious. Last time I comment this. 


#76
Dec1310, 07:09 PM

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#77
Dec1310, 07:11 PM

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#78
Dec1310, 07:38 PM

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Chalnoth, as you previously teach me some stuff I should not refuse to answer you.



#79
Dec1310, 08:01 PM

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Baloney. Busy beavers halt. Halting is not a requirement of a Turing machine program. An algorithm to compute the decimal representation of an irrational computable numbers such as the square root of 2 will not stop. The decimal representation of course requires an infinitelylong tape.



#80
Dec1310, 08:15 PM

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We are getting a little off topic. But.
1. A turing machine must have access to an unlimited amount of tape in order to capture the full power of the turing machine formalism. 2. Any given turing machine, when run on an input for which it terminates, will only use a finite amount of that tape. The lack of a limitation on space can't be waved away by saying "but that's only for terminating programs", because a significant portion of the power of turing machines in the first place comes from the fact that turing machines have the option of never terminating. The decidable languages are a smaller set than the recognizable languages. If there is a known bound on either runtime or space used, then all programs are "decidable" and you have something weaker than a turing machine. You are correct that the busy beaver function does define an upper bound on space usage for halting programs. However this statement is not useful to know, because BB(n) is noncomputable. (In other words, BB(n) is a bound, but it is an unknown and unknowable bound). In fact your statement here is basically tautological, because BB(n) is defined as the maximum number of tape squares used up by an eventuallyterminating program of a given complexity. So what you are saying is that the number of squares used by a terminating program will be equal to or less than the number of squares it uses. Do you disagree? 


#81
Dec1310, 09:06 PM

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I think a theory of quantum gravity would probably be a TOE, because it would unit QM with GR, particles and fields with spacetime. And to unite spacetime and particles fields would probably require a theory from reason alone. For physically, there is nothing more fundamental than spacetime and particles/fields. And to explain something more fundamental than what is physical would have to rely on complete, abstract, generality about anything true; it would have to rely on principle alone. So here we are discussing whether physics can be reduced to a complete axiomatic system. The question is not even relevant unless we can derive physics from a system of reasoning. I don't think it will reduce to the axioms of geometry because we would still have to explain why there is geometry, or spacetime, to begin with. It would have to reduce to reason, or we would still have questions and it would not be complete. (am I using "complete" equivocally here?) 


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Dec1310, 09:12 PM

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#83
Dec1310, 09:12 PM

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#84
Dec1310, 09:35 PM

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#85
Dec1310, 10:49 PM

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#86
Dec1310, 10:58 PM

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Originally Posted by friend
It would have to reduce to reason, or we would still have questions and it would not be complete. (am I using "complete" equivocally here?) I'm not so sure. "reduce to reason" is a reference to logic which is relevant to Godel's Theorems. "still have questions" asks about how the subject in question is proven, which goes to Godel's incompleteness theorem. So it might be my use of the word "complete" is on topic. 


#87
Dec1310, 11:08 PM

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#88
Dec1310, 11:16 PM

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#89
Dec1310, 11:17 PM

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I'll sketch the idea, but it's going from memory so, as ever, I could be wrong. If you like, think of Hilbert's second order axioms as being replaced with schemas, plus axioms asserting the existence of first order definable regions to give us the points to give us the lines, planes and regions of the theory (I disagree with Hurkyl that these regions exist as a matter of first order logic from the theory of points). Then one can define a region R which consists of an infinite sequence of points p1, p2, p3..., such that all points lie on the same line, p1p2 Cong p2p3, p2p3Congp3p4...., there is no point p on R such that p1 lies between p2 and p.... Basically, we can define the existence of a region using the points of an arbitrarily, infinite spaced region with one endpoint, and these then behave like the natural numbers. Certain geometric constructions allow us, within this theory, to define addition and multiplication on this points of this region, and then we have enough material to do the Godel construction. 


#90
Dec1310, 11:27 PM

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Your response still sounds to me like you're suggesting that reality is not completely logical. I don't think that is the view of science. I think that we go off in search of answers because we think things are rational. I don't think that any credible scientist is trying to prove that reality is illogical. 


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