Tom Caylor wrote:
> OK.  I noticed that you can get the Universal Machine (UM) to run for
> ever even without the "+ 1".  If I think of the program for G as a big
> "case statement" with cases 1, 2, 3, to infinity, then the case for k
> will contain the code for, or better yet a call to (hence the name
> "recursive"?), Fk(k), but if we state by defining even G = Fn(n) (even
> without the "+ 1") then this is equivalent to calling G(k)...  But then
> when we call G(k) we end up back in the "k case" again, calling G(k)
> again,... forever.  This will happen even if we add the "+ 1".
> Personally I like this argument (running forever) better than the 0 = 1
> argument that somehow concludes that the UM will crash.  A UM
> "crashing" to me brings up pictures of physical machines that recognize
> an unallowed operation, and then stop themselves.
>

And on the surface, it seems that the "running forever because of
self-reference" argument is better because you don't need the "+ 1".
It seems that it isn't the "+ 1" that makes the UM run forever, and
conversely the UM runs forever even without the contradiction of 0 = 1.

Tom


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