Re: An All/Nothing multiverse model
At 17:15 03/12/04 -0500, Hal Ruhl wrote: Hi Bruno: I assume your theory is intended to give the range of descriptions of worlds. The All in my model contains - well - ALL so it includes systems to which Godel's theorem applies. Your theory has problems for me. What is truth? Truth is a queen who wins all the wars without any army. You can guess it by reading a newspaper. But you can better guess it by reading two independent newspaper, and still better by reading three independent newspapers, etc. What is a sentence? An informal sentence is a ordered set of words having hopefully some meaning. A formal sentence is the same but with a decidable grammar, and sometimes a mathematical notion of meaning in the form of a mathematical structure satisfying the sentence. This can be find in any textbook in logic. What is arithmetical? A sentence is arithmetical, roughly, if it bears on (natural) numbers. As Stephen Paul King asked: How is truth resolved for a given sentence? It is resolved partially by proof. Why the down select re descriptions vs the All. I don't understand. How is the set of such sentences known to be consistent? It is never known to be consistent. We can just hope it is. (Smullyan makes a different case for arithmetical truth, but this would be in contradiction with the comp hyp). To answer these questions it seems necessary to inject information into your theory beyond what may already be there - the sentences - ... Right. This indeed follows from Goedel's incompleteness. ...and where did all that info come from and why allow any in a base level system for worlds? Concerning just natural numbers this is a mystery. With comp it is necessarily mysterious. Best regards, Bruno http://iridia.ulb.ac.be/~marchal/
Re: An All/Nothing multiverse model
Hi Bruno: In my questions about truth etc I was not really looking for a response but was rather trying to demonstrate the need for additional information in your theory. Your responses made my point I think. It is this issue I struggle with. I seek a TOE that has no net information. Though its components individually may have any amount of information the sum of all the information in all the components is no information. At 08:13 AM 12/6/2004, you wrote: At 17:15 03/12/04 -0500, Hal Ruhl wrote: Hi Bruno: I assume your theory is intended to give the range of descriptions of worlds. The All in my model contains - well - ALL so it includes systems to which Godel's theorem applies. Your theory has problems for me. What is truth? Truth is a queen who wins all the wars without any army. You can guess it by reading a newspaper. But you can better guess it by reading two independent newspaper, and still better by reading three independent newspapers, etc. What is a sentence? An informal sentence is a ordered set of words having hopefully some meaning. A formal sentence is the same but with a decidable grammar, and sometimes a mathematical notion of meaning in the form of a mathematical structure satisfying the sentence. This can be find in any textbook in logic. What is arithmetical? A sentence is arithmetical, roughly, if it bears on (natural) numbers. As Stephen Paul King asked: How is truth resolved for a given sentence? It is resolved partially by proof. Why the down select re descriptions vs the All. I don't understand. My theory almost [However see below] includes yours as a sub component. My only spin is that my theory necessarily has all dynamics in it subject to external random input. Why down select to just your theory and as a result add all that extra required info? How is the set of such sentences known to be consistent? It is never known to be consistent. We can just hope it is. That is what I thought. (Smullyan makes a different case for arithmetical truth, but this would be in contradiction with the comp hyp). Please give me a URL or reference for his work. To answer these questions it seems necessary to inject information into your theory beyond what may already be there - the sentences - ... Right. This indeed follows from Goedel's incompleteness. Here you appear to me to be saying that your theory is indeed subject to random external input. Random because we do not know if the set of sentences is consistent in its current state and if incomplete it can be added to. How can it be added to in a manner that is consistent with the existing state? . So it would seem that your theory is indeed a sub component of my theory so as I said why down select and be burdened with all that net info? ...and where did all that info come from and why allow any in a base level system for worlds? Concerning just natural numbers this is a mystery. With comp it is necessarily mysterious. Perhaps it is mysterious because it is unnecessary. Hal
Re: An All/Nothing multiverse model
Hal Ruhl wrote: To answer these questions it seems necessary to inject information into your theory beyond what may already be there - the sentences - ... Right. This indeed follows from Goedel's incompleteness. Here you appear to me to be saying that your theory is indeed subject to random external input. Random because we do not know if the set of sentences is consistent in its current state and if incomplete it can be added to. How can it be added to in a manner that is consistent with the existing state? . We can choose whether a Godel statement should be judged true or false by consulting our model of arithmetic. See this post of mine on the use of models in mathematics from the thread Something for Platonists (you can see the other posts in the thread by clicking 'View This Thread' at the top): http://www.escribe.com/science/theory/m4584.html Jesse
Re: An All/Nothing multiverse model
Hi Jesse: My originating post appeals only to the result of Turing to the effect that there is in general no decision procedure. As a result FAS in general can not be both complete and consistent. Since my All contains all FAS including the complete ones then the All is inconsistent. That is the simplicity of it. As to any confusion over the concept of model I can call just as well call it a theory. Hal At 02:40 PM 12/6/2004, you wrote: Hal Ruhl wrote: To answer these questions it seems necessary to inject information into your theory beyond what may already be there - the sentences - ... Right. This indeed follows from Goedel's incompleteness. Here you appear to me to be saying that your theory is indeed subject to random external input. Random because we do not know if the set of sentences is consistent in its current state and if incomplete it can be added to. How can it be added to in a manner that is consistent with the existing state? . We can choose whether a Godel statement should be judged true or false by consulting our model of arithmetic. See this post of mine on the use of models in mathematics from the thread Something for Platonists (you can see the other posts in the thread by clicking 'View This Thread' at the top): http://www.escribe.com/science/theory/m4584.html Jesse
Re: An All/Nothing multiverse model
Hal Ruhl wrote: Hi Jesse: My originating post appeals only to the result of Turing to the effect that there is in general no decision procedure. There's no single decision procedure for a Turing machine, but if you consider more general kinds of machines, like a hypercomputer that can check an infinite number of cases in a finite time, then there may be a single decision procedure for such a machine to decide if any possible statement about arithmetic is true or false. If your everything includes only computable universes, then such hypercomputers wouldn't exist in any universe, but if you believe in an everything more like Tegmark's collection of all conceivable mathematical structures, then there should be universes where it would be possible to construct such a hypercomputer, even if they can't be constructed in ours. By the way, do you understand that Godel's proof is based on the idea that, if we have an axiomatic system A, we can always find a statement G that we can understand to mean axiomatic system A will not prove statement G to be true? Surely it is not simply a matter of random choice whether G is true or false--we can see that as long as axiomatic system A is consistent, it cannot prove G to be false (because that would mean axiomatic system A [i]will[/i] prove G to be true), nor can it prove it is true (because that would mean it was proving true the statement that it would never prove it true). But this means that A will never prove G true, which means we know G *is* true, provided A is consistent. I would say that we can *know* that the Peano axioms are consistent by consulting our model of arithmetic, in the same way we can *know* the axiomatic system discussed in my post at http://www.escribe.com/science/theory/m4584.html is consistent, by realizing those axioms describe the edges and vertices of a triangle. Do you disagree that these model-based proofs of consistency are valid? Jesse
Re: An All/Nothing multiverse model
Hi Jesse: I think you miss my point. The All contains ALL including Turing machines that model complete FAS and other inconsistent systems. The All is inconsistent - that is all that is required. Godel's theorem is a corollary of Turing's. As you say a key element of Godel's approach to incompleteness is to assume consistency of the system in question. The only way I see to falsify my theory at this location is to show that all contents of the All are consistent. Hal At 11:46 PM 12/6/2004, you wrote: Hal Ruhl wrote: Hi Jesse: My originating post appeals only to the result of Turing to the effect that there is in general no decision procedure. There's no single decision procedure for a Turing machine, but if you consider more general kinds of machines, like a hypercomputer that can check an infinite number of cases in a finite time, then there may be a single decision procedure for such a machine to decide if any possible statement about arithmetic is true or false. If your everything includes only computable universes, then such hypercomputers wouldn't exist in any universe, but if you believe in an everything more like Tegmark's collection of all conceivable mathematical structures, then there should be universes where it would be possible to construct such a hypercomputer, even if they can't be constructed in ours. By the way, do you understand that Godel's proof is based on the idea that, if we have an axiomatic system A, we can always find a statement G that we can understand to mean axiomatic system A will not prove statement G to be true? Surely it is not simply a matter of random choice whether G is true or false--we can see that as long as axiomatic system A is consistent, it cannot prove G to be false (because that would mean axiomatic system A [i]will[/i] prove G to be true), nor can it prove it is true (because that would mean it was proving true the statement that it would never prove it true). But this means that A will never prove G true, which means we know G *is* true, provided A is consistent. I would say that we can *know* that the Peano axioms are consistent by consulting our model of arithmetic, in the same way we can *know* the axiomatic system discussed in my post at http://www.escribe.com/science/theory/m4584.html is consistent, by realizing those axioms describe the edges and vertices of a triangle. Do you disagree that these model-based proofs of consistency are valid? Jesse
Re: An All/Nothing multiverse model
Hal Ruhl wrote: Hi Jesse: I think you miss my point. The All contains ALL including Turing machines that model complete FAS and other inconsistent systems. The All is inconsistent - that is all that is required. You mean because the All contains Turing machines which model axiomatic systems that are provably inconsistent (like a system that contains the axiom all A have property B as well as the axiom there exists an A that does not have property B), that proves the All itself is inconsistent? If that's your argument, I don't think it makes sense--the Turing machine itself won't behave in a contradictory way as it prints out symbols, there will always be a single definite truth about which single it prints at a given time, it's only when we interpret the *meaning* of those symbols that we may see the machine has printed out two symbol-strings with opposite meaning. But we are free to simply believe that the machine has printed out a false statement, there is no need to believe that every axiomatic system describes an actual world within the All, even a logically impossible world where two contradictory statements are simultaneously true. Godel's theorem is a corollary of Turing's. As you say a key element of Godel's approach to incompleteness is to assume consistency of the system in question. But do you agree it is possible for us to *prove* the consistency of a system like the Peano arithmetic or the axiomatic system describing the edges and points of a triangle, by finding a model for the axioms? The only way I see to falsify my theory at this location is to show that all contents of the All are consistent. Hal I think you need to give a more clear definition of what is encompassed by the All before we can decide if it is consistent or inconsistent. For example, does the All represent the set of all logically possible worlds, or do you demand that it contains logically impossible worlds too? Does the All contain sets of truths that cannot be printed out by a single Turing machine, but which could be printed out by a program written for some type of hypercomputer, like the set of all true statements about arithmetic (a set which is both complete and consistent)? Jesse
