Hi Bruno and everybody,
>> I >> hope to send my comments and/or 'OK' sign :-) on Monday. > > Take it easy. There is no deadline on the list. Making a declaration helps me to get things done. Yet I'm late. Whenever you see such sentences in my posts, you can skip it, they are mostly for me :-) --------------- I'll try to write a summary in my own words. Let's see how much I did understand. Prepositions: A .... finite alphabet A* .... finite words over A (it is enumerable, moreover effectively enumerable) L ..... a language over A. E .... a subset of A*, a set of valid expressions in L (obviously, it is at least enumerable) M ..... a machine which understands to L f ..... a total function from N to N. Goal: I want to develop a universal language L which describes all and only all functions f. Given an expression from E, M computes the result in finite time. Given the restrictions on L, the result is a number and nothing else. The set of all functions f from N to N is not enumerable (by Cantor's diagonal). Thus there is no bijection to E. Thus, I have to find a smaller set of functions. I will call this set a set of computable functions, C. Inevitably, this is computability by definition, by definition of L. So L should be 'really good' in order to encompass as much functions f from N to N as possible. Now, I think of a bijection between E and C. 0 --- E_0 ~ f_0 1 --- E_1 ~ f_1 2 --- E_2 ~ f_2 3 --- E_3 ~ f_3 .... .... Since E are efficiently enumerable, C are efficiently enumerable as well. Yes, it might happen that f_i = f_j for i != j, but is does not matter as long as all unique f_i are in the enumeration. Time for the Kleene diagonal argument. Opps, a language L that I dreamt of does not exist. I have to relax from the condition that M on E_i always return a number in a finite time. Well, what to return if not a number ... nothing -> M experiences an infinite loop. What a world, ok, my language has to describe total functions from N to N and as well as strict partial functions from N to N. And it is clear that I cannot know whether E_i corresponds to a total function or a strict partial function. f' stands for any function descriable by L. 0 --- E_0 ~ f'_0 1 --- E_1 ~ f'_1 2 --- E_2 ~ f'_2 3 --- E_3 ~ f'_3 .... .... N, E and C are enumerable, moreover obviously effectively enumerable. Any subset of C is at least enumerable. A subset of C corresponding to total functions is no effectively enumerable. It cannot be. Well, a language L which describes both total and strict partial functions and hereby defines a set of computable functions is not refuted by Kleene's diagonal. It is not obvious to me how strong sort of evidence that universal L exists, it is to survive Kleene's diagnonal. Droping an apple on the floor is in favor of Newton's law but not very convincing :-) Oh, now I realize that my question is actually weird. Since the set of computable functions is defined by L, and L is said to be universal if it describes exactly these functions - it is simple to develop a trivial L -> it defines a set of computable functions ... and of course universal L exists. In this sence a universal language L always exists. So I write it rather like this: If we develop many sufficiently different programming languages and they turn out to be all equivalent, it might convince me that the set of 'computable functions' is fixed. Although, written like this I can think of educated (math) people who will tell me: This is all you have???? So, what are like 2-3 most direct consequences of CT which make CT to seem rock-solid? Here I assume that CT basically says that the set of functions descriable by the lambda calculus is all what I can ever compute. ------------- Regarding points 3)-5) of your summary, I am lost on terms such as Absolute ontic TOE, Observer Moments, Aristotelian principles, Machine theology, ... ------------- I wrote down a list of short-term goals on what I would like to have some background/knowledge with a help from this list: 1\ I saw somewhere a sentence saying approximately this: ".... so universe is performing a computation. Is then universe a big computer? No." I would like to know in a broad sense what it tries to say a why one shoud rather accept it or reject. 2\ Bruno, you recently wrote that you do not agree with Wolfram's Principle of Computational Equivalence. As I understand to that principle, Wolfram says that universe is a big cellular automata. What is the evidence that it is unlikely this way? Sincerely, Mirek --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---
