Last post before the key post (was OM = SIGMA_1) 1bis
Mirek, Le 28-nov.-07, à 17:32, Mirek Dobsicek a écrit : Hi Bruno, I'm ready. Luckily, it is not long time ago, I've received my university degree in CS, so it was rather easy to follow :-) Sincerely, Mirek Thanks for telling me that you are ready. Now I feel a bit guilty because today and tomorrow I get unexpected work, and next week I am teaching again. I hope that those who have no university degree in CS have been able to follow the thread too. I will try to resume the last exercise tomorrow, (one last post on Cantor's diagonal), and then, I will write, during next week, the key post, which will prove an absolutely fundamental theorem on the Universal Machines, a theorem without which UDA would be stuck in the sixth step, and without which the lobian interview would not make sense. The theorem says that ALL universal machines are insecure or imperfect. I guess some of you can already guess or produce the proof (in company of a general definition of secure machine, 'course). Torgny, You should be clearer about when you work *in* your ultrafinistic theory and when you work in its metatheory. If not, Quentin is right to ask you not to mention any sort of infinite of any kind. Most of the time, it is very hard to make sense of your approach, due to the lack of a clear distinction between the ultrafinistic theory and the informal metatheory you do refer to, very often. Note that without the movie graph (the 8th step of the UDA), comp remains coherent *only* through an explicitly physicalist version of ultrafinitism and an explicitly dualist theory of Mind (perhaps you should collaborate with Marc?). Mind would need matter (but then why, and what is it?), and the UDA would not go through because we would live in a unique and then very little universe. I guess everythingers would be skeptical at the start, here. Also the quantum facts are going in an opposite direction, imo. Actually, the movie-graph prevents such a move, I think. We can go back on this, later. To be sure I am open to critics there, I am not entirely satisfied with my presentation of the argument, and both George and Russell did succeed in making me thinking a lot more on that issue, or of the way to present it perhaps (more than I was expecting). Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Theory of Everything based on E8 by Garrett Lisi
Quentin Anciaux skrev: Hi, Le Wednesday 28 November 2007 09:56:17 Torgny Tholerus, vous avez écrit : You only need models of cellular automata. If you have a model and rules for that model, then one event will follow after another event, according to the rules. And after that event will follow another more event, and so on unlimited. The events will follow after eachother even if you will not have any implementation of this model. Any physics is not needed. You don't need any geometric properties. Sure, but you can't be ultrafinitist and saying things like And after that event will follow another more event, and so on unlimited. There is a difference between unlimited and infinite. Unlimited just says that it has no limit, but everything is still finite. If you add something to a finite set, then the new set will always be finite. It is not possible to create an infinite set. So it is OK to use the word unlimited. But it is not OK to use the word infinite. Is this clear? Another important word is the word all. You can talk about all events. But in that case the number of events will be finite, and you can then talk about the last event. But you can't deduce any contradiction from that, because that is forbidden by the type theory. And there will be more events after the last event, because the number of events is unlimited. As soon as you use the word all, you will introduce a limit - all up to this limit. And you must then think of only doing conclusions that are legal according to type theory. So the best thing is to avoid the word all (and all synonyms of that word). -- Torgny --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Theory of Everything based on E8 by Garrett Lisi
Le Thursday 29 November 2007 17:22:59 Torgny Tholerus, vous avez écrit : Quentin Anciaux skrev: Hi, Le Wednesday 28 November 2007 09:56:17 Torgny Tholerus, vous avez écrit : You only need models of cellular automata. If you have a model and rules for that model, then one event will follow after another event, according to the rules. And after that event will follow another more event, and so on unlimited. The events will follow after eachother even if you will not have any implementation of this model. Any physics is not needed. You don't need any geometric properties. Sure, but you can't be ultrafinitist and saying things like And after that event will follow another more event, and so on unlimited. There is a difference between unlimited and infinite. Unlimited just says that it has no limit, but everything is still finite. If you add something to a finite set, then the new set will always be finite. It is not possible to create an infinite set. I'm sorry I don't get it... The set N as an infinite numbers of elements still every element in the set is finite. Maybe it is an english subtility that I'm not aware of... but in french I don't see a clear difference between infini and illimité. So it is OK to use the word unlimited. But it is not OK to use the word infinite. Is this clear? No, I don't see how a set which have not limit get a finite number of elements. Another important word is the word all. You can talk about all events. But in that case the number of events will be finite, and you can then talk about the last event. But you can't deduce any contradiction from that, because that is forbidden by the type theory. And there will be more events after the last event, because the number of events is unlimited. If there are events after the last one, how can the last one be the last ? As soon as you use the word all, you will introduce a limit - all up to this limit. And you must then think of only doing conclusions that are legal according to type theory. o_O... could you explain what is type theory ? So the best thing is to avoid the word all (and all synonyms of that word). like everything ? Regards, Quentin Anciaux -- All those moments will be lost in time, like tears in the rain. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Theory of Everything based on E8 by Garrett Lisi
Quentin Anciaux skrev: Le Thursday 29 November 2007 17:22:59 Torgny Tholerus, vous avez écrit : There is a difference between unlimited and infinite. Unlimited just says that it has no limit, but everything is still finite. If you add something to a finite set, then the new set will always be finite. It is not possible to create an infinite set. I'm sorry I don't get it... The set N as an infinite numbers of elements still every element in the set is finite. Maybe it is an english subtility that I'm not aware of... but in french I don't see a clear difference between infini and illimité. As soon as you talk about the set N, then you are making a closure and making that set finite. The only possible way to talk about something without limit, such as natural numbers, is to give a production rule, so that you can produce as many of that type of objects as you want. If you have a natural number n, then you can produce a new number n+1, that is the successor of n. So it is OK to use the word unlimited. But it is not OK to use the word infinite. Is this clear? No, I don't see how a set which have not limit get a finite number of elements. It is not possible for a set to have no limit. As soon as you construct a set, then that set will always have a limit. Either you have to accept that the set N is finite, or you must stop talking about the set N. It is enough to have a production rule for natural numbers. Another important word is the word all. You can talk about all events. But in that case the number of events will be finite, and you can then talk about the last event. But you can't deduce any contradiction from that, because that is forbidden by the type theory. And there will be more events after the last event, because the number of events is unlimited. If there are events after the last one, how can the last one be the last ? The last event is the last event in the set of all events. But because you have a production rule for the events, it is always possible to produce new events after the last event. But these events do not belong to the set of all events. As soon as you use the word all, you will introduce a limit - all up to this limit. And you must then think of only doing conclusions that are legal according to type theory. o_O... could you explain what is type theory ? Type theory is one of the solutions of Russel's paradox. You have a hierarchy of types. Type theory says that the all quantifiers only can span objects of the same type (or lower types). When you create new objects, such that the set of all sets that do not belong to themselves, then you will get an object of a higher type, so that you can not say anything about if this set belongs to itself or not. The same thing with the set of all sets. You can not say anything about if it belongs to itself. So the best thing is to avoid the word all (and all synonyms of that word). like everything ? Yes... :-) -- Torgny --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Theory of Everything based on E8 by Garrett Lisi
Le Thursday 29 November 2007 18:25:54 Torgny Tholerus, vous avez écrit : Quentin Anciaux skrev: Le Thursday 29 November 2007 17:22:59 Torgny Tholerus, vous avez écrit : There is a difference between unlimited and infinite. Unlimited just says that it has no limit, but everything is still finite. If you add something to a finite set, then the new set will always be finite. It is not possible to create an infinite set. I'm sorry I don't get it... The set N as an infinite numbers of elements still every element in the set is finite. Maybe it is an english subtility that I'm not aware of... but in french I don't see a clear difference between infini and illimité. As soon as you talk about the set N, then you are making a closure and making that set finite. Ok then the set R is also finite ? The only possible way to talk about something without limit, such as natural numbers, is to give a production rule, so that you can produce as many of that type of objects as you want. If you have a natural number n, then you can produce a new number n+1, that is the successor of n. What is the production rules of the noset R ? So it is OK to use the word unlimited. But it is not OK to use the word infinite. Is this clear? No, I don't see how a set which have not limit get a finite number of elements. It is not possible for a set to have no limit. As soon as you construct a set, then that set will always have a limit. I don't get it. Either you have to accept that the set N is finite, or you must stop talking about the set N. It is enough to have a production rule for natural numbers. I don't accept and/or don't understand. Another important word is the word all. You can talk about all events. But in that case the number of events will be finite, and you can then talk about the last event. But you can't deduce any contradiction from that, because that is forbidden by the type theory. And there will be more events after the last event, because the number of events is unlimited. If there are events after the last one, how can the last one be the last ? The last event is the last event in the set of all events. But because you have a production rule for the events, it is always possible to produce new events after the last event. But these events do not belong to the set of all events. There exists no last element in the set N. As soon as you use the word all, you will introduce a limit - all up to this limit. And you must then think of only doing conclusions that are legal according to type theory. o_O... could you explain what is type theory ? Type theory is one of the solutions of Russel's paradox. You have a hierarchy of types. Type theory says that the all quantifiers only can span objects of the same type (or lower types). When you create new objects, such that the set of all sets that do not belong to themselves, then you will get an object of a higher type, so that you can not say anything about if this set belongs to itself or not. The same thing with the set of all sets. You can not say anything about if it belongs to itself. So the best thing is to avoid the word all (and all synonyms of that word). like everything ? Yes... :-) What you are saying seems like to me So the best thing is to avoid words at all (and any languages)... Regards, Quentin Anciaux -- All those moments will be lost in time, like tears in the rain. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Theory of Everything based on E8 by Garrett Lisi
Quentin Anciaux skrev: Le Thursday 29 November 2007 18:25:54 Torgny Tholerus, vous avez écrit : As soon as you talk about the set N, then you are making a closure and making that set finite. Ok then the set R is also finite ? Yes. The only possible way to talk about something without limit, such as natural numbers, is to give a production rule, so that you can produce as many of that type of objects as you want. If you have a natural number n, then you can produce a new number n+1, that is the successor of n. What is the production rules of the noset R ? How do you define the set R? -- Torgny --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Theory of Everything based on E8 by Garrett Lisi
Le Thursday 29 November 2007 18:52:36 Torgny Tholerus, vous avez écrit : Quentin Anciaux skrev: Le Thursday 29 November 2007 18:25:54 Torgny Tholerus, vous avez écrit : As soon as you talk about the set N, then you are making a closure and making that set finite. Ok then the set R is also finite ? Yes. o_O The only possible way to talk about something without limit, such as natural numbers, is to give a production rule, so that you can produce as many of that type of objects as you want. If you have a natural number n, then you can produce a new number n+1, that is the successor of n. What is the production rules of the noset R ? How do you define the set R? http://en.wikipedia.org/wiki/Construction_of_real_numbers Choose your method... Regards, Quentin Anciaux -- All those moments will be lost in time, like tears in the rain. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
RE: Theory of Everything based on E8 by Garrett Lisi
Date: Thu, 29 Nov 2007 18:25:54 +0100 From: [EMAIL PROTECTED] To: [EMAIL PROTECTED] Subject: Re: Theory of Everything based on E8 by Garrett Lisi Quentin Anciaux skrev: Le Thursday 29 November 2007 17:22:59 Torgny Tholerus, vous avez écrit : There is a difference between unlimited and infinite. Unlimited just says that it has no limit, but everything is still finite. If you add something to a finite set, then the new set will always be finite. It is not possible to create an infinite set. I'm sorry I don't get it... The set N as an infinite numbers of elements still every element in the set is finite. Maybe it is an english subtility that I'm not aware of... but in french I don't see a clear difference between infini and illimité. As soon as you talk about the set N, then you are making a closure and making that set finite. Why is that? How do you define the word set? The only possible way to talk about something without limit, such as natural numbers, is to give a production rule, so that you can produce as many of that type of objects as you want. If you have a natural number n, then you can produce a new number n+1, that is the successor of n. Why can't I say the set of all numbers which can be generated by that production ruler? It almost makes sense to say a set is *nothing more* than a criterion for deciding whether something is a member of not, although you would need to refine this definition to deal with problems like Russell's set of all sets that are not members of themselves (which could be translated as the criterion, 'any criterion which does not match its own criterion'--I suppose the problem is that this criterion is not sufficiently well-defined to decide whether it matches its own criterion or not). So it is OK to use the word unlimited. But it is not OK to use the word infinite. Is this clear? No, I don't see how a set which have not limit get a finite number of elements. It is not possible for a set to have no limit. As soon as you construct a set, then that set will always have a limit. Is there something intrinsic to your concept of the word set that makes this true? Is your concept of a set fundamentally different than my concept of well-defined criteria for deciding if any given object is a member or not? Jesse --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Theory of Everything based on E8 by Garrett Lisi
Quentin Anciaux skrev: Le Thursday 29 November 2007 18:52:36 Torgny Tholerus, vous avez écrit : Quentin Anciaux skrev: What is the production rules of the noset R ? How do you define the set R? http://en.wikipedia.org/wiki/Construction_of_real_numbers Choose your method... The most important part of that definition is: 4. The order ? is /complete/ in the following sense: every non-empty subset of *R* bounded above http://en.wikipedia.org/wiki/Upper_bound has a least upper bound http://en.wikipedia.org/wiki/Least_upper_bound. This definition can be translated to: If you have a production rule that produces rational numbers that are bounded above, then this production rule is producing a real number. This is the production rule for real numbers. -- Torgny --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Theory of Everything based on E8 by Garrett Lisi
Jesse Mazer skrev: From: [EMAIL PROTECTED] As soon as you talk about the set N, then you are making a closure and making that set finite. Why is that? How do you define the word set? The only possible way to talk about something without limit, such as natural numbers, is to give a production rule, so that you can produce as many of that type of objects as you want. If you have a natural number n, then you can produce a new number n+1, that is the successor of n. Why can't I say the set of all numbers which can be generated by that production ruler? As soon as you say the set of ALL numbers, then you are forced to define the word ALL here. And for every definition, you are forced to introduce a limit. It is not possible to define the word ALL without introducing a limit. (Or making an illegal circular definition...) It almost makes sense to say a set is *nothing more* than a criterion for deciding whether something is a member of not, although you would need to refine this definition to deal with problems like Russell's set of all sets that are not members of themselves (which could be translated as the criterion, 'any criterion which does not match its own criterion'--I suppose the problem is that this criterion is not sufficiently well-defined to decide whether it matches its own criterion or not). A well-defined criterion is the same as what I call a production rule. So you can use that, as long as the criterion is well-defined. (What does the criterion, that decides if an object n is a natural number, look like?) It is not possible for a set to have no limit. As soon as you construct a set, then that set will always have a limit. Is there something intrinsic to your concept of the word set that makes this true? Is your concept of a set fundamentally different than my concept of well-defined criteria for deciding if any given object is a member or not? Yes, the definition of the word all is intrinsic in the concept of the word set. -- Torgny --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
RE: Theory of Everything based on E8 by Garrett Lisi
Date: Thu, 29 Nov 2007 19:55:20 +0100 From: [EMAIL PROTECTED] To: [EMAIL PROTECTED] Subject: Re: Theory of Everything based on E8 by Garrett Lisi Jesse Mazer skrev: From: [EMAIL PROTECTED] As soon as you talk about the set N, then you are making a closure and making that set finite. Why is that? How do you define the word set? The only possible way to talk about something without limit, such as natural numbers, is to give a production rule, so that you can produce as many of that type of objects as you want. If you have a natural number n, then you can produce a new number n+1, that is the successor of n. Why can't I say the set of all numbers which can be generated by that production ruler? As soon as you say the set of ALL numbers, then you are forced to define the word ALL here. And for every definition, you are forced to introduce a limit. It is not possible to define the word ALL without introducing a limit. (Or making an illegal circular definition...) Why can't you say If it can be generated by the production rule/fits the criterion, then it's a member of the set? I haven't used the word all there, and I don't see any circularity either. It almost makes sense to say a set is *nothing more* than a criterion for deciding whether something is a member of not, although you would need to refine this definition to deal with problems like Russell's set of all sets that are not members of themselves (which could be translated as the criterion, 'any criterion which does not match its own criterion'--I suppose the problem is that this criterion is not sufficiently well-defined to decide whether it matches its own criterion or not). A well-defined criterion is the same as what I call a production rule. So you can use that, as long as the criterion is well-defined. (What does the criterion, that decides if an object n is a natural number, look like?) I would just define the criterion recursively by saying 1 is a natural number, and given a natural number n, n+1 is also a natural number. Jesse --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
RE: Bijections (was OM = SIGMA1)
Date: Tue, 20 Nov 2007 19:01:38 +0100
From: [EMAIL PROTECTED]
To: [EMAIL PROTECTED]
Subject: Re: Bijections (was OM = SIGMA1)
Bruno Marchal skrev:
But infinite ordinals can be different, and still have the same
cardinality. I have given examples: You can put an infinity of linear
well founded order on the set N = {0, 1, 2, 3, ...}.
The usual order give the ordinal omega = {0, 1, 2, 3, ...}. Now omega+1
is the set of all ordinal strictly lesser than omega+1, with the
convention above. This gives {0, 1, 2, 3, ... omega} = {0, 1, 2, 3, 4,
{0, 1, 2, 3, 4, }}. As an order, and thus as an ordinal, it is
different than omega or N. But as a cardinal omega and omega+1 are
identical, that means (by definition of cardinal) there is a bijection
between omega and omega+1. Indeed, between {0, 1, 2, 3, ... omega} and
{0, 1, 2, 3, ...}, you can build the bijection:
0omega
10
21
32
...
n --- n-1
...
All right?- represents a rope.
An ultrafinitist comment:
In the last line of this sequence you will have:
? - omega-1
But what will the ? be? It can not be omega, because omega is not
included in N...
--
Torgny
There is no such ordinal as omega-1 in conventional mathematics. Keep in mind
that ordinals are always defined as sets of previous ordinals, with 0 usually
defined as the empty set {}...So,
0 = {}
1 = {0} = {{}}
2 = {0, 1} = {{}, {{}}}
3 = {0, 1, 2} = {{}, {{}}, {{}, {{
...and so forth. In thes terms, the ordinal omega is the set of finite
ordinals, or:
omega = {0, 1, 2, 3, 4, ... } = too much trouble for me to write out in brackets
How would the set omega-1 be defined? It doesn't make sense unless you
believe in a last finite ordinal, which of course a non-ultrafinitist will
not believe in.
Jesse
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Re: Theory of Everything based on E8 by Garrett Lisi
Marc, please, allow me to write in plain language - not using those fancy words of these threads. Some time ago when the discussion was in commonsensically more understandable vocabulary, I questioned something similar to Günther, as pertaining to numbers - the alleged generators of 'everything' (physical, quality, ideation, process, you name it). As Bruno then said: the positive integers do that - if applied in sufficiently long expressions. (please, Bruno, correct this to a bottom-low simplification) - I did not follow that and was promised some more explanatory text in not so technical language. The discussion over the past some weeks is even more technical for me. Is not the distinction relevant what I hold, that there are two kinds of 'number'-usage: the (pure, theoretical Math and the in sciences - (quantity related) - applied math - that uses the formalism (the results, even logics) of 'Math' to exercise 'math'? (Cap vs lower m) Geometry seems to be in between() and symmetry can be both, I think. I am no physicist AND no mathematician, (not even a logician), so I pretend to keep an objective eye on things in which I am not prejudiced by knowledge. (G). John M On Nov 27, 2007 11:40 PM, [EMAIL PROTECTED] wrote: On Nov 28, 1:18 am, Günther Greindl [EMAIL PROTECTED] wrote: Dear Marc, Physics deals with symmetries, forces and fields. Mathematics deals with data types, relations and sets/categories. I'm no physicist, so please correct me but IMHO: Symmetries = relations Forces - could they not be seen as certain invariances, thus also relating to symmetries? Fields - the aggregate of forces on all spacetime points - do not see why this should not be mathematical relation? The mathemtical entities are informational. The physical properties are geometric. Geometric properties cannot be derived from informational properties. Why not? Do you have a counterexample? Regards, Günther Don't get me wrong. I don't doubt that all physical things can be *described* by mathematics. But this alone does not establish that physical things *are* mathematical. As I understand it, for the examples you've given, what happens is that based on emprical observation, certain primatives of geometry and symmetry are *attached to* (connected with) mathematical relations, numbers etc which successfully *describe/predict* these physical properties. But it does not follow from this, that the mathematical relations/numbers *are* the geometric properties/symmetrics. In order to show that the physical properties *are* the mathematical properties (and not just described by or connected to the physical properties), it has to be shown how geometric/physical properties emerge from/are logically derived from sets/categories/numbers alone. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
