At 07:56 -0600 1998/11/03, Randolph J. Herber wrote: >I disagree with your second point---with sufficient encoding, any >semantic can be preserved, possibly with a time penalty (which are >ignored when discussing such equivalences). That is one of the >points of Goedel's Incompleteness (Undecidability) Theorem. Provided the semantics that one wants to describe can be expressed by binary numbers: For example, TeX proper cannot draw a non-straight spline curve no matter how you apply Turing theorems and Godel theorem, even though you can define structures simulating that TeX can handle that -- but that has no practical significance unless you figure out a way to extend TeX to print it. (In this example, even though the curve itself can be described by binary numbers, the capacity of printing it cannot.) Hans Aberg * Email: Hans Aberg <mailto:[log in to unmask]> * Home Page: <http://www.matematik.su.se/~haberg/> * AMS member listing: <http://www.ams.org/cml/>