Re: An All/Nothing multiverse model
From: Hal Ruhl <[EMAIL PROTECTED]> To: [EMAIL PROTECTED] Subject: Re: An All/Nothing multiverse model Date: Tue, 07 Dec 2004 23:22:40 -0500 Hi Jesse: The All contains inconsistent FAS [we have no issue here as far as I can tell] I'm not so sure--if your "All" does not include deterministic Turing machine computations, but only "states" of Turing machines which are visited randomly, then it seems to me that the All should not include axiomatic systems which deterministically output a series of theorems either--in analogy with isolated Turing machine states, it should just contain individual isolated theorems, and (according to your theory) visit different theorems at random. Unless by the "state" of a Turing machine you mean its final endstate after it has finished the computation, in which case maybe this could be analogous to the final set of *all* theorems that can ever be proved by a given FAS. and thus all of the theorems of such FAS as some of the kernels of information simultaneously. [Do we have an issue here?] Are you saying a "kernel of information" is a set of all possible theorems that a given FAS can prove? This content makes the All inconsistent. [OK?] No, I still don't understand in what sense you think the All is inconsistent, but if you can explain in concrete terms what you mean by "kernels of information" perhaps I would see what your argument is. Jesse
Re: An All/Nothing multiverse model
From: Hal Ruhl <[EMAIL PROTECTED]> To: [EMAIL PROTECTED] Subject: Re: An All/Nothing multiverse model Date: Tue, 07 Dec 2004 22:41:45 -0500 Maybe this will help: The All contains all possible output states of all Turing machines [among all manner of other info such as states of really messy universes] simultaneously. These states are given "Physical reality" by evolving Somethings in random order over and over. Some such sequences can arbitrarily closely approach or even exactly match those that would be output by a Turing machine for long runs of states [but not infinite runs of states due to the random input factor - no selection allowed]. All other sequences of all kinds of states also take place. Hal OK, that is helpful in making your ideas a little more concrete. But in this case, what would it mean for two possible states to be "inconsistent" with one another? Can you give an example of two Turing machine states that are inconsistent? Also, when you talk about Turing machine states, are you talking about different possible strings of numbers on the tape that will be seen *after* a given Turing machine's computation has halted, or are you talking about the state of a Turing machine during a single step in its computation, like "the tape reads 100011010, the Turing machine's read/write head is on the second zero, and the machine is in internal state #14"? Jesse
Re: An All/Nothing multiverse model
Hi Jesse: The All contains inconsistent FAS [we have no issue here as far as I can tell] and thus all of the theorems of such FAS as some of the kernels of information simultaneously. [Do we have an issue here?] This content makes the All inconsistent. [OK?] The All does not output anything - it is internally inconsistent. [OK?]. A Something [see the original post] can not evolve [its boundary moving through the All in an attempt to complete itself ] consistent with its prior evolution because each new kernel encompassed by its boundary changes the Something and further some such kernels may be inconsistent with those kernels already encompassed. [OK?] Further the consistent evolution of a Something would be a selection [evolution according to some plan] which is not allowed [see original post] [OK?] This in no way prevents any kind of string of states from being encompassed. [OK?] Hal
Re: An All/Nothing multiverse model
From: Hal Ruhl <[EMAIL PROTECTED]> To: "Jesse Mazer" <[EMAIL PROTECTED]> Subject: Re: An All/Nothing multiverse model Date: Tue, 07 Dec 2004 22:19:02 -0500 Hi Jesse: At 09:23 PM 12/7/2004, you wrote: Hal Ruhl wrote: To clarify - the All contains all information simultaneously [see the definition in the original post] - including ALL Truing machines with ALL possible output tapes - so it contains simultaneously both output tapes re your comment below. But if there is a fact which is true in one "world" being simulated by a given Turing machine, but false in a different Turing machine simulation, that doesn't mean that "the All" is contradictory. After all, the statement "this planet contains life" is true of Earth but not true of Pluto, but that doesn't mean the solar system is contradictory, it just means that different facts are true of different planets. This really misses my meaning. That is not how Somethings evolve in the All. The Somethings incorporate preexisting information such as states of universes in a random dynamic. I am not asking about how "Somethings" evolve in your theory, I'm asking what's your justification for claiming that the All is inconsistent. You are giving examples of machines simulating worlds. That is not how my approach works. Thus my response. For the other see below. As long as you always describe the *context* of any statement, I don't see any reason why we should describe the All as inconsistent. So if you think the All is inconsistent somehow, you need to explain in more detail why you think this is. I already have. Would you agree that Turing's result says that some subset of FAS are inconsistent? You don't need Turing's results to show that, Its one of many ways of showing that the All contains kernels of information that are inconsistent with each other. The kernels are always there. No computers are running in my All it only may look that way here and there from time to time. What is a "kernel of information"? Can you give a concrete example of two kernels of information within the All that are inconsistent with each other? However, there is a distinction between saying an axiomatic system is inconsistent, and saying there is something inconsistent in the behavior of the Turing machine simulating that system. There will always be a single definite truth about what symbol the Turing machine prints out at what time--it is only when you try to interpret the *meaning* of different strings of symbols that it prints out that you will see an inconsistency. As an analogy, suppose I am running a complex simulation of a human being sitting at a writing desk, and he writes two sentences on a simulated piece of paper: "I have a beard" and "I do not have a beard". If we interpret these sentences in terms of their english meaning, obviously they represent inconsistent statements, but that doesn't mean the simulation itself is somehow "inconsistent", does it? One of the statements will be true and one will be false, so there's no problem. Get rid of the machine. OK, instead of talking about a simulated person running on a machine, let's just talk about a "real person" like you or me, whatever you think real people are. If I write the words "I have a beard" and then write the words "I do not have a beard", does this show the All is inconsistent? If not, then why does the fact that we can write down (or conceive of) inconsistent axiomatic systems show that the All is inconsistent? Your argument would only show the All to be inconsistent if you believe that for every axiomatic system a Turing machine can simulate, there must be a corresponding "world" within the All where all the axioms and theorems represent simultaneously true statements about that world. But if you believe that, then you are saying the All must contain not only all possible worlds, but logically impossible worlds as well. Is that what you're saying? All states of all worlds are logically within the venue and visited with "physical reality" over and over. What is "the venue"? Can you give an example of what you mean by a "state" of a world? Can you explain why the fact that there are inconsistent axiomatic systems shows that All is inconsistent? Also, you didn't answer my earlier question about whether your idea of the All only includes worlds that could be simulated on a Turing machine, or if it also includes worlds that could be simulated by a "hypercomputer" which is capable of performing uncomputable operations (like instantly deciding if a given Turing machine program will halt or not). The All is all information without restriction. All the information is in there all the time. The boundaries of the Somethings wash across the inherent counterfactuals counterfactually. I don't understand what these words are supposed to mean, or how they address my question above. Can you just answer "yes" or "no"? Again get rid of the machine. The dynamic
Re: An All/Nothing multiverse model
Maybe this will help: The All contains all possible output states of all Turing machines [among all manner of other info such as states of really messy universes] simultaneously. These states are given "Physical reality" by evolving Somethings in random order over and over. Some such sequences can arbitrarily closely approach or even exactly match those that would be output by a Turing machine for long runs of states [but not infinite runs of states due to the random input factor - no selection allowed]. All other sequences of all kinds of states also take place. Hal
Re: An All/Nothing multiverse model
Hi Jesse: At 09:23 PM 12/7/2004, you wrote: Hal Ruhl wrote: To clarify - the All contains all information simultaneously [see the definition in the original post] - including ALL Truing machines with ALL possible output tapes - so it contains simultaneously both output tapes re your comment below. But if there is a fact which is true in one "world" being simulated by a given Turing machine, but false in a different Turing machine simulation, that doesn't mean that "the All" is contradictory. After all, the statement "this planet contains life" is true of Earth but not true of Pluto, but that doesn't mean the solar system is contradictory, it just means that different facts are true of different planets. This really misses my meaning. That is not how Somethings evolve in the All. The Somethings incorporate preexisting information such as states of universes in a random dynamic. I am not asking about how "Somethings" evolve in your theory, I'm asking what's your justification for claiming that the All is inconsistent. You are giving examples of machines simulating worlds. That is not how my approach works. Thus my response. For the other see below. As long as you always describe the *context* of any statement, I don't see any reason why we should describe the All as inconsistent. So if you think the All is inconsistent somehow, you need to explain in more detail why you think this is. I already have. Would you agree that Turing's result says that some subset of FAS are inconsistent? You don't need Turing's results to show that, Its one of many ways of showing that the All contains kernels of information that are inconsistent with each other. The kernels are always there. No computers are running in my All it only may look that way here and there from time to time. it is quite trivial to construct an axiomatic system with two contradictory axioms, or with different subsets of axioms that can be used to prove inconsistent theorems. However, there is a distinction between saying an axiomatic system is inconsistent, and saying there is something inconsistent in the behavior of the Turing machine simulating that system. There will always be a single definite truth about what symbol the Turing machine prints out at what time--it is only when you try to interpret the *meaning* of different strings of symbols that it prints out that you will see an inconsistency. As an analogy, suppose I am running a complex simulation of a human being sitting at a writing desk, and he writes two sentences on a simulated piece of paper: "I have a beard" and "I do not have a beard". If we interpret these sentences in terms of their english meaning, obviously they represent inconsistent statements, but that doesn't mean the simulation itself is somehow "inconsistent", does it? One of the statements will be true and one will be false, so there's no problem. Get rid of the machine. Your argument would only show the All to be inconsistent if you believe that for every axiomatic system a Turing machine can simulate, there must be a corresponding "world" within the All where all the axioms and theorems represent simultaneously true statements about that world. But if you believe that, then you are saying the All must contain not only all possible worlds, but logically impossible worlds as well. Is that what you're saying? All states of all worlds are logically within the venue and visited with "physical reality" over and over. Also, you didn't answer my earlier question about whether your idea of the All only includes worlds that could be simulated on a Turing machine, or if it also includes worlds that could be simulated by a "hypercomputer" which is capable of performing uncomputable operations (like instantly deciding if a given Turing machine program will halt or not). The All is all information without restriction. All the information is in there all the time. The boundaries of the Somethings wash across the inherent counterfactuals counterfactually. I don't understand what these words are supposed to mean, or how they address my question above. Can you just answer "yes" or "no"? Again get rid of the machine. The dynamic is not a simulation generating states in any way. Hal
Re: An All/Nothing multiverse model
At 06:37 PM 12/7/2004, you wrote: To clarify - the All contains all information simultaneously [see the definition in the original post] - including ALL Truing machines with ALL possible output tapes - so it contains simultaneously both output tapes re your comment below. But if there is a fact which is true in one "world" being simulated by a given Turing machine, but false in a different Turing machine simulation, that doesn't mean that "the All" is contradictory. After all, the statement "this planet contains life" is true of Earth but not true of Pluto, but that doesn't mean the solar system is contradictory, it just means that different facts are true of different planets. This really misses my meaning. That is not how Somethings evolve in the All. The Somethings incorporate preexisting information such as states of universes in a random dynamic. Similarly, if the All contains all "possible worlds" in some sense (all possible Turing machine programs, for example), then different facts could be true of different worlds, without this meaning the All itself is inconsistent. If Turing machine program #2334 simulates a 3-dimensional universe while Turing machine program #716482 simulates a 2-dimensional universe, that doesn't mean the inconsistent statements "the universe is 3-dimensional" and "the universe is 2-dimensional" are simultaneously true in the All--rather, it just means the statements "the universe described by program #2334 is 3-dimensional" and "the universe described by program #716482 is 2-dimensional" are simultaneously true in the All, and there is no contradiction between these statements. See above. As long as you always describe the *context* of any statement, I don't see any reason why we should describe the All as inconsistent. So if you think the All is inconsistent somehow, you need to explain in more detail why you think this is. I already have. Would you agree that Turing's result says that some subset of FAS are inconsistent? Also, you didn't answer my earlier question about whether your idea of the All only includes worlds that could be simulated on a Turing machine, or if it also includes worlds that could be simulated by a "hypercomputer" which is capable of performing uncomputable operations (like instantly deciding if a given Turing machine program will halt or not). The All is all information without restriction. All the information is in there all the time. The boundaries of the Somethings wash across the inherent counterfactuals counterfactually. Hal
Re: An All/Nothing multiverse model
Hal Ruhl wrote: To clarify - the All contains all information simultaneously [see the definition in the original post] - including ALL Truing machines with ALL possible output tapes - so it contains simultaneously both output tapes re your comment below. But if there is a fact which is true in one "world" being simulated by a given Turing machine, but false in a different Turing machine simulation, that doesn't mean that "the All" is contradictory. After all, the statement "this planet contains life" is true of Earth but not true of Pluto, but that doesn't mean the solar system is contradictory, it just means that different facts are true of different planets. This really misses my meaning. That is not how Somethings evolve in the All. The Somethings incorporate preexisting information such as states of universes in a random dynamic. I am not asking about how "Somethings" evolve in your theory, I'm asking what's your justification for claiming that the All is inconsistent. As long as you always describe the *context* of any statement, I don't see any reason why we should describe the All as inconsistent. So if you think the All is inconsistent somehow, you need to explain in more detail why you think this is. I already have. Would you agree that Turing's result says that some subset of FAS are inconsistent? You don't need Turing's results to show that, it is quite trivial to construct an axiomatic system with two contradictory axioms, or with different subsets of axioms that can be used to prove inconsistent theorems. However, there is a distinction between saying an axiomatic system is inconsistent, and saying there is something inconsistent in the behavior of the Turing machine simulating that system. There will always be a single definite truth about what symbol the Turing machine prints out at what time--it is only when you try to interpret the *meaning* of different strings of symbols that it prints out that you will see an inconsistency. As an analogy, suppose I am running a complex simulation of a human being sitting at a writing desk, and he writes two sentences on a simulated piece of paper: "I have a beard" and "I do not have a beard". If we interpret these sentences in terms of their english meaning, obviously they represent inconsistent statements, but that doesn't mean the simulation itself is somehow "inconsistent", does it? One of the statements will be true and one will be false, so there's no problem. Your argument would only show the All to be inconsistent if you believe that for every axiomatic system a Turing machine can simulate, there must be a corresponding "world" within the All where all the axioms and theorems represent simultaneously true statements about that world. But if you believe that, then you are saying the All must contain not only all possible worlds, but logically impossible worlds as well. Is that what you're saying? Also, you didn't answer my earlier question about whether your idea of the All only includes worlds that could be simulated on a Turing machine, or if it also includes worlds that could be simulated by a "hypercomputer" which is capable of performing uncomputable operations (like instantly deciding if a given Turing machine program will halt or not). The All is all information without restriction. All the information is in there all the time. The boundaries of the Somethings wash across the inherent counterfactuals counterfactually. I don't understand what these words are supposed to mean, or how they address my question above. Can you just answer "yes" or "no"? Jesse
Re: An All/Nothing multiverse model
Hi Bruno: At 06:40 AM 12/7/2004, you wrote: Hi Hal, 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. I don't have a theory. Just an argument showing that if we are machine then eventually physics is derivable from machine psychology/computer science. I have almost no current opposition to this. It sounds to me that it is in the All with my adder of a random input to the machine. 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. 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. I deduce this from many readings of Smullyan. But I think Smullyan is just afraid that people takes Godel's second incompleteness theorem as an argument showing that Peano Arithmetic cannot been known to be consistent. And I agree with Smullyan on that point. I believe we discussed this and you agreed that a complete arithmetic would be inconsistent. I have not found the applicable posts. But with comp I cannot know my own consistency and I can only show (to myself) that IF I am consistent then Peano Arithmetic is consistent. Look at the "Forever Undecided" book (on the net or in the list archive). There seems to be many ways to establish the necessary and sufficient properties of my All and the above seems to be one of them. 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. Not the theory, but the possible observers described by theory. This is just a consequence of comp: we "belongs' to an uncountable infinity of (infinite) computations. Cf our talk on the white rabbits. We don't need to inject randomness: a priori we have too much (first person) randomness. With comp it is the *lack* of randomness which is in need to be explained. The randomness injected at each event can be quite small. Also it is injected into each Something which itself is a multiverse so it is spread over all the universes in that multiverse. Seldom would it parse so as to inject large deltas into individual universes. "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? This is not relevant. See Jesse's post. But not wrong? See my previous post which is a clearer statement of what I mean. The above is a contribuitor to the random evolution dynamic of the Somethings. Two identical Somethings may not take the same next step. 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? But which theory? COMP ? COMP is mainly the hope that it is possible to survive some treatment in a hospital. We have reached too many levels of nesting. I have been of on my own excavations. Is not "all true arithmetical sentences" a part of comp? ...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. But then you should explain why we believe in natural numbers. (You did give plenty evidence that you believe in natural numbers). They would be in the All. Hal
Re: An All/Nothing multiverse model
From: Hal Ruhl <[EMAIL PROTECTED]> To: [EMAIL PROTECTED] Subject: Re: An All/Nothing multiverse model Date: Tue, 07 Dec 2004 10:46:04 -0500 Hi Jesse: To clarify - the All contains all information simultaneously [see the definition in the original post] - including ALL Truing machines with ALL possible output tapes - so it contains simultaneously both output tapes re your comment below. But if there is a fact which is true in one "world" being simulated by a given Turing machine, but false in a different Turing machine simulation, that doesn't mean that "the All" is contradictory. After all, the statement "this planet contains life" is true of Earth but not true of Pluto, but that doesn't mean the solar system is contradictory, it just means that different facts are true of different planets. Similarly, if the All contains all "possible worlds" in some sense (all possible Turing machine programs, for example), then different facts could be true of different worlds, without this meaning the All itself is inconsistent. If Turing machine program #2334 simulates a 3-dimensional universe while Turing machine program #716482 simulates a 2-dimensional universe, that doesn't mean the inconsistent statements "the universe is 3-dimensional" and "the universe is 2-dimensional" are simultaneously true in the All--rather, it just means the statements "the universe described by program #2334 is 3-dimensional" and "the universe described by program #716482 is 2-dimensional" are simultaneously true in the All, and there is no contradiction between these statements. As long as you always describe the *context* of any statement, I don't see any reason why we should describe the All as inconsistent. So if you think the All is inconsistent somehow, you need to explain in more detail why you think this is. Also, you didn't answer my earlier question about whether your idea of the All only includes worlds that could be simulated on a Turing machine, or if it also includes worlds that could be simulated by a "hypercomputer" which is capable of performing uncomputable operations (like instantly deciding if a given Turing machine program will halt or not). Jesse
Re: An All/Nothing multiverse model
Hi Jesse: To clarify - the All contains all information simultaneously [see the definition in the original post] - including ALL Truing machines with ALL possible output tapes - so it contains simultaneously both output tapes re your comment below. It is not a time dependent or belief dependent issue. If one could go fishing in the All as an evolving Something essentially does - you would eventually pull out both tapes in random order just like the order in which someone catches a big fish or a little fish. The fish and the fisherman are also in no fixed relation - no selection. The boundary defining a given Something moves through the All and will encompass these various tapes in no fixed order - no selection - it is random input to that Something. Once a Something incorporates a particular kernel of information its boundary necessarily moves according to that total content - it is a new Something and it is a journey towards completion for that configuration. The fisherman catches the big fish and goes home happy never catching the little fish, or, or , etc., etc. The boundary of each Something takes an unknown and unknowable [random] path. Here all states of universes are encompassed [the instant of "physical reality"] again and again. Some [most I suppose] states can be quite messy but so what? They are logically possible within the venue as are neat ones. However, long long strings of neat ones absent large deltas between the states that are given "physical reality" and having small deltas that are "reasonable" happen. The idea that some of these strings of states could be simulated on a computer is also in the All but the computer must have one port that allows random input. Hal At 01:49 AM 12/7/2004, you wrote: 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
Re: An All/Nothing multiverse model
Hi Hal, 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. I don't have a theory. Just an argument showing that if we are machine then eventually physics is derivable from machine psychology/computer science. 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. 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. I deduce this from many readings of Smullyan. But I think Smullyan is just afraid that people takes Godel's second incompleteness theorem as an argument showing that Peano Arithmetic cannot been known to be consistent. And I agree with Smullyan on that point. But with comp I cannot know my own consistency and I can only show (to myself) that IF I am consistent then Peano Arithmetic is consistent. Look at the "Forever Undecided" book (on the net or in the list archive). 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. Not the theory, but the possible observers described by theory. This is just a consequence of comp: we "belongs' to an uncountable infinity of (infinite) computations. Cf our talk on the white rabbits. We don't need to inject randomness: a priori we have too much (first person) randomness. With comp it is the *lack* of randomness which is in need to be explained. "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? This is not relevant. See Jesse's post. 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? But which theory? COMP ? COMP is mainly the hope that it is possible to survive some treatment in a hospital. ...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. But then you should explain why we believe in natural numbers. (You did give plenty evidence that you believe in natural numbers). Bruno http://iridia.ulb.ac.be/~marchal/
