>> But you get what I'm saying. Maybe there is some fundamentally new 
>> system that changes the rules, so to speak.
> 
> That still won't solve the halting problem, because the halting problem 
> isn't defined in terms of this fundamentally new system.
> 
> It's like saying "integers are not closed under division", and then 
> saying "but we've discovered rationals!"   Integers *still* aren't 
> closed under division, even if you invent rationals.
> 
>> Even if such a system were to exist, you would still have a new, 
>> generalised Halting Problem, and you're back to square one.
> 
> That's rather harder to say, really, since we by definition have no idea 
> what a program in this entirely new paradigm would look like. But I 
> suspect you're correct.

The idea is that if you had some super-machine that could somehow decide 
whether any given Turing machine halts for a given input, you would 
still be unable to tell whether this brand new machine halts for a given 
input - using exactly the same proof as the original halting problem.

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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