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.

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?




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