On 13 Dec 2009, at 18:20, Jason Resch wrote:

> On Sun, Dec 13, 2009 at 10:25 AM, Bruno Marchal <marc...@ulb.ac.be>  
> wrote:
> >
> > Though in another way I think we already have a theory of  
> everything a
> > theory can explain *ultimately* (which is *not even remotely*  
> close to
> > everything, since the more you trascend a theory the "bigger" the
> > possibilities get):
> > The theory is that all theories are either contradictory or
> > incomplete (we
> > have to go beyond theories to access truth). I think Gödel already
> > made the
> > quest for the "complete" theory meaningless.
> Gödel showed that all theories on *numbers* are contradictory of
> incomplete.
> And it is a direct consequence of Church thesis. Once you grasp the
> concept of universal number or machine, you understand that truth,
> even on just machines and numbers, is not completely axiomatisable.
> But that is a reason to be humble  in front of arithmetical truth. Not
> a reason to dismiss it. It kicks back a lot.
> Also, if you mention Gödel, it means you accept elementary arithmetic.
> My logical point is that if you believe you (can) surivive with a
> digital *body*, then elementary arithmetic has to be enough. WE have
> too extract the SWE, and other appearances from that. It is a point in
> (applied) logic, if you want.
>  Bruno,
> I have had some difficulty in seeing how to get from the numbers and  
> arithmetic to universal machines and programs such as the universal  
> dovetailer.  For example, the existence of the Java language doesn't  
> directly imply all possible Java programs are being executed  
> somewhere.  Is there some example you can provide of how to get from  
> numbers to the execution of programs?  I've been thinking about it  
> myself for a while and this is the closest I have gotten, is it  
> along the right track?
> 1. If all natural numbers exist, then relations between those  
> numbers exist (e.g. 5 is 3 more than 2)
> 2. There are an infinite number of ways to get from some number x to  
> number y (e.g. if x is 2, and y is 5: y = x^2+1, y = x + 3, y = x *  
> 3 - 1) are all valid relations between 2 and 5.
> 3. Every relation, may be applied recursively to generate an  
> infinite sequence of numbers, the simplest relation: y=x+1, when  
> applied recursively gives all the successors, others more complex  
> ones might give the Fibonacci sequence, or run through states of the  
> Game of Life.
> Is this enough?  It seems like something is being added on top of  
> the numbers, the relations themselves must be treated as independent  
> entities, as well as recursively applied relations for every  
> number.  Is there a simpler or more obvious way the existence of  
> numbers yields the dovetailer?

It is a long and tedious exercise to show that the computable  
relations can be represented in the form of arithmetical relations  
(provable in an already rather weak theory).

I have defined computations by sequences of phi_i^s(n) for s = 0, 1,  
3, 4, .... Those sequences can be represented in first order  
arithmetic, and the relevant one to describe the universal dovetailer  
can be represented as well and proved (by weak theories).

Good question, though. I will think how to explain this more  
explicitly later, but not too much because it is usually longer than  
programing an operating system in language machine. A big part of that  
work is what Gödel did in his incompleteness proof: to represent  
"metamathemetical notion" in arithmetic. Like provability can be  
translated in arithmetic, concept like universal machine and  
computations can also be translated. this needs a rather long  
"explanation", given that the machine (or elementary arithmetic) a  
priori knows nothing about those notions.




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