Hi Kim, Hi Marty and others, So it is perhaps time to do some math. Obviously, once we are open to the idea that the fundamental reality could be mathematical, it is normal to take some time to do some mathematics. Many people seems also to agree here that the computationalist hypothesis could be interesting, and this should motivate for some amount of theoretical computer science, or recursion theory. This is a branch of mathematics which study computability, and mainly uncomputability, as opposed to computation theory which study all aspect of computation.

You can easily show that something is computable, by giving an explicit procedure to compute it. But to show that something is NOT computable, you need a very solid notion of computability. Now, computationalism suggest that the interesting and fundamental things, like life, consciousness, even matter, "lives" somehow on the border of the computable and the non computable, so that it is perhaps time to dig a little bit deeper in those direction. I have already explain this on this list, but never from scratch, having in mind those who are, for whatever reason, the mathematical basis. So I guess that many of view will find those preliminaries a bit too much simple. yet, by experience, I know that difficulties will appear, and it is frequent that I met people with very big baggages in mathematics who have some difficulties to understand the final point. So I would encourage everyone to be sure everything is clear. For those who have already a thorough understanding of UDA1-7, and in particular have grasped the difference between a computation and a description of a computation, I ask them to just be patient with the list. There will be nothing new here, nothing really original, and nothing controversial. All what I will explain has been anticipate by Emil Post in the 1920, and found or rediscovered by many mathematicians independently in USA, and in the ex USSR. The present post just give a preview, and the beginning . I intent to send only short posts, or the shorter as possible. So here is the plan. 1) Set (probably a dozen of posts) 2) Function (I don't know how many posts, could depend on the replies. It is the key notion of math) 3) Language, machine and computable function 4) Universal machine, universal language, universal function, universal dovetailer, universal number ... How will I proceed? By exercise only. I will ask question, and I will wait for either the answer. I expect that those "who know" wait for Kim's answer, or for answer by those who are not supposed to "know". Kim seems to courageously accept the role of the candid, but if someone else want to answer it is OK for me. I will give only VERY SIMPLE exercise. The goal is to be short and simple, and as informal as possible. I will explain by examples, and I will avoid as much as possible tedious definitions. And when we will meet a more difficult question, I will just solve it myself, and even explain why it is difficult. So, those preliminaries will not be very funny. I will of course adapt myself to the possible replies, and fell free to comment or even metacomment. Obviously this is a not a course in math, but it is an explanation from scratch of the seven step of the universal dovetailer argument. It is a shortcut, and most probably we will make some digression from time to time, but let us try not to digress too much. Kim, you are OK with this? I have to take into account the problem you did have with math, and which makes this lesson a bit challenging for me, and I guess for you too. I begin with the very useful and elementary notion of set, as explained in what is called "naive set theory", and which is the base of almost all part of math. ============================================= begin =============================== 1) SET Informal definition: a set is a collection of object, called elements, with the idea that it, the collection or set, can be considered itself as an object. It is a many seen as a one, if you want. If the set is not to big, we can describe it exhaustively by listing the elements, if the set is bigger, we can describe it by some other way. Usually we use accolades "{", followed by the elements, separated by commas, and then "}", in the exhaustive description of a set. Example/exercise: 1) The set of odd natural numbers which are little than 10. This is a well defined, and not to big set, so we can describe it exhaustively by {1, 3, 5, 7, 9}. In this case we say that 7 belongs to {1, 3, 5, 7, 9}. Exercise 1: does the number 24 belongs to the set {1, 3, 5, 7, 9}? 2) the set of even natural number which are little than 13. It is {0, 2, 4, 6, 8, 10, 12}. OK? Some people can have a difficulty which is not related to the notion of set, for example they can ask themselves if zero (0) is really an even number. We will come back to this. 3) The set of odd natural numbers which are little than 100. This set is already too big to describe exhaustively. We will freely describe such a set by a quasi exhaustion like {1, 3, 5, 7, 9, 11, ... 95, 97, 99}. Exercise 2: does the number 93 belongs to the set of odd natural numbers which are little than 100, that is: does 93 belongs to {1, 3, 5, 7, 9, 11, ... 95, 97, 99}? 4) The set of all natural numbers. This set is hard to define, yet I hope you agree we can describe it by the infinite quasi exhaustion by {0, 1, 2, 3, ...}. Exercise 3: does the number 666 belongs to the set of natural numbers, that is does 666 belongs to {0, 1, 2, 3, ...}. Exercice 4: does the real number square-root(2) belongs to {0, 1, 2, 3, ...}? 5) When a set is too big or cumbersome, mathematician like to give them a name. They will usually say: let S be the set {14, 345, 78}. Then we can say that 14 belongs to S, for example. Exercise 5: does 345 belongs to S? A set is entirely defined by its elements. Put in another way, we will say that two sets are equal if they have the same elements. Exercise 6. Let S be the set {0, 1, 45} and let M be the set described by {45, 0, 1}. Is it true or false that S is equal to M? Exercise 7. Let S be the set {666} and M be the set {6, 6, 6}. Is is true or false that S is equal to M? Seven exercises are enough. Are you ready to answer them. I hope you don't find them too much easy, because I intend to proceed in a way such that all exercise will be as easy, despite we will climb toward very much deeper notion. Feel free to ask question, comments, etc. I will try to adapt myself. Next: we will see some operation on sets (union, intersection), and the notion of subset. If all this work, I will build a latex document, and make it the standard reference for the seventh step for the non mathematician, or for the beginners in mathematics. 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 everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---