  We introduce some early considerations of physical and mathematical
impossibility as preludes to the Goedel incompleteness theorems. We consider
some informal aspects of these theorems and their underlying assumptions and
discuss some the responses to these theorems by those seeking to draw
conclusions from them about the completability of theories of physics. We argue
that there is no reason to expect Goedel incompleteness to handicap the search
for a description of the laws of Nature, but we do expect it to limit what we
can predict about the outcomes of those laws, and examples are given. We
discuss the Goedel universe and the role it played in exposing the full
spectrum of possibilities that a global understanding of space-time would
reveal. Finally,we show how recent studies of supertasks have shown how global
space-time structure determines the ultimate capability of computational
devices within them.
