### Re: Combined response Re: Computing Randomness

Hal Ruhl wrote: I appreciate the conversation so I will try to build a common reference so each additional step to my model can be built on that base and individually commented on. As requested these are definitions and terms relevant to my model not necessarily to established mathematics.

### Re: Combined response Re: Computing Randomness

Hal Ruhl wrote: I appreciate the conversation so I will try to build a common reference so each additional step to my model can be built on that base and individually commented on. As requested these are definitions and terms relevant to my model not necessarily to established mathematics.

### Re: Combined response Re: Computing Randomness

Dear Bruno: Thank you for your patience and the excellent response. You should try to make your model part of established mathematics. Not for the glory, but for making it comprehensible. That is what I am trying to do here, but since I have proven to have too few current mathematical skills

### Re: Combined response Re: Computing Randomness

Dear Bruno: I appreciate the conversation so I will try to build a common reference so each additional step to my model can be built on that base and individually commented on. As requested these are definitions and terms relevant to my model not necessarily to established mathematics. 1a)

### Re: Combined response Re: Computing Randomness

Dear Bruno: At , you wrote: Hal Ruhl wrote: The assumption leads to a contradiction when String N exceeds the complexity allowed by Chaitin. More information must be added to the cascade for it to continue. Why ? Only if your FAS produces as output just the string N and then stop, then

### Combined response Re: Computing Randomness

Dear Juergen and Bruno: Clearly I have a problem when I try to use mathematical terminology in which I am not formally trained to explain my approach. So here is an attempt to explain it in just a few normal words. My system be it a FAS or not is modeled on the logistics equation process

### Re: Computing Randomness

Hal, I think you really might want to read some introductory textbook on logic and formal systems, to check the standard definitions of `proof' and `theorem.' BTW, the following remarkable method heavily depends on what's provable. I believe it will find its way into general computer science

### Re: Late response to Bruno Re: Computing Randomness

Hal Ruhl wrote: In what sense 4+1= is a proof chain ? A proof must be a sequence of formula each of which are either axiom instance or theorems. IMO it is ... Definitions are not matter of opinion, but of conventional consensus. ... a sequence of: 1) A formula which in this case is of

### Late response to Bruno Re: Computing Randomness

Dear Bruno: Sorry I missed this. Here is my response. At , you wrote: Hal Ruhl wrote: Juergen: Hal, here is an infinite chain of provable unique theorems: 1+1=2, 2+1=3, 3+1=4, 4+1=5, ... First these are not theorems they are proof chains ending in theorems. If you reinterpret

### Re: Computing Randomness

Dear Hal Let me give my view one more try. An N-bit FAS is N-bits because that is its compressed form. I see it as nothing but a bag of elegant proofs. There are only 2^(N + c) that can fit in the bag. Hal

### Re: Computing Randomness

Dear Hal Here is the second quote. It is from Chaitin's The Limits of Mathematics page 90. The first of these theorems states that an N-bit formal axiomatic system cannot enable one to exhibit any specific object with a program-size complexity greater than N + c. Hal

### Re: Computing Randomness

Hal writes: Here is a direct quote from page 24 of Chaitin's The Unknowable: The general flavor of my work is like this. You compare the complexity of the axioms to the complexity of the result you're trying to derive, and if the result is more complex than the axioms, then you can not get

### Re: Computing Randomness

Hal Ruhl wrote: Juergen: Hal, here is an infinite chain of provable unique theorems: 1+1=2, 2+1=3, 3+1=4, 4+1=5, ... First these are not theorems they are proof chains ending in theorems. If you reinterpret Juergen's word then you can tell him anything. In all presentations of

### Re: Computing Randomness

Dear Hal Since I was previously convinced by another that side bar discussions should be avoided I will respond to this on the list. At 4/12/01, you wrote: Hal writes: You are writing programs and they have a complexity. Chaitin limits this complexity to no more than the complexity of the

### Re: Computing Randomness

Hal writes: Well any assertion [object] with a LISP elegant program size greater than N + 356 can not be fully described by A since you can not identify its elegant program with A. Agreed. Now Chaitin says on page 24 that he can not exhibit specific true, unprovable assertions. But

### Re: Computing Randomness

Dear Juergen: In case what I tried to say was not clear the idea is that there are no more than 2^(N + c) shortest possible unique proofs in an N-bit FAS. How can number theory if it is a finite FAS contain an infinite number of unique theorems? Hal

### Re: Computing Randomness

Dear Jacques: At 4/12/01, you wrote: Maybe Hal, Russel and Jurgen should take this discussion to email and just let us know how it turns out, because I get enough junk mail already. I have run into those who do not like the side bar approach. I tend to agree that it cuts all the others

### Re: Computing Randomness

Dear Juergen: You demonstrate my point. At 4/12/01, you wrote: Hal, here is an infinite chain of provable unique theorems: 1+1=2, 2+1=3, 3+1=4, 4+1=5, ... First these are not theorems they are proof chains ending in theorems. For example: 4 + 1 = is a proof chain and the theorem proved is: 5

### Re: Computing Randomness

Dear Russell: Yes we did indeed have a similar debate some time ago. At that time I was still trying to express this point of view correctly and admittedly made a number of mistakes back then [and still do]. Our debate helped me considerably and I thank you. In response: I just posted a

### Re: Computing Randomness

Let me try and short circuit this debate, since I had precisely this debate with Hal about 18 months ago, where I found myself in the same position Juergen finds himself now. Basically, Hal believes a finite FAS by definition implies that each theorem is constrained to be no more than N-bits in

### Re: Computing Randomness

Dear Russell: You wrote: Why bound the proof? It was not my idea. Chaitin equated complexity with a computing program's length and a proof chain is a computing program according to Turing. [rearranging your post] 1+1=2, 2+1=3, 3+1=4 ... are all distinct theorems. My view: Again as in my

### Re: Computing Randomness

Dear Russell: At 4/13/01, you wrote: Bounded complexity does not imply bounded length. Examples include an infinite sting of '0's, and the string '1234...9101112...' That was part of the old debate and one of my initial mistakes. I am not now talking about the length of theorems but the

### Re: Computing Randomness

Maybe Hal, Russel and Jurgen should take this discussion to email and just let us know how it turns out, because I get enough junk mail already. From: Hal Ruhl [EMAIL PROTECTED] T4=T1 and T2, T5=T1 or T2, T6=T1 and T2 or T3, are also theorems. We can construct an infinite variety of these

### Re: Computing Randomness

Hal Ruhl wrote: Dear Russell: At 4/13/01, you wrote: Bounded complexity does not imply bounded length. Examples include an infinite sting of '0's, and the string '1234...9101112...' That was part of the old debate and one of my initial mistakes. I am not now talking about the length

### Re: Computing Randomness

Dear Juergen: At 4/11/01, you wrote: Hal, Chaitin just says you cannot prove 20 pound theorems with 10 pound axioms. Please refer to Chaitin's The Unknowable generally and page 25, Chapter V, and note 10 at the bottom of page 97 in particular. But the infinite cascade of all provable theorems

### Re: Computing Randomness

Dear Juergen: For the cite to Chaitin see The Limits to Mathematics page 17 note 1 at the bottom of the page. Hal

### Re: Computing Randomness

Hal, you wrote: I believe that attempting an extensive detailed formal description of the Everything is the wrong approach. IMO - at least in this case - the more information used to describe, the smaller the thing described. I was not able to follow this, but informal and vague descriptions

### Re: Computing Randomness

I appreciate Juergen's view. In essence he is assuming a nonuniform distribution on the ensemble of descriptions, as though the ensemble of descriptions are produced by the FAST algorithm. This is perhaps the same as assuming a concrete universe. In my approach (which didn't have this technical

### RE: Computing Randomness

Dear Juergen: In reply: Where does all the randomness come from? Many physicists would be content with a statistical theory of everything (TOE) based on simple probabilistic physical laws allowing for stochastic predictions such as We do not know where this particular electron will be in the