In this chapter, I introduce some early considerations of physical and mathematical impossibility as preludes to Gödel's incompleteness theorems. I consider some informal aspects of these theorems and their underlying assumptions and discuss some of the responses to these theorems by those seeking to draw conclusions from them about the completability of theories of physics. Also, I argue that there is no reason for us to expect Gödel 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 I have provided some examples. I then discuss the “Gödel universe” – a solution to Einstein's equations describing a rotating universe where time travel is possible, as demonstrated by Gödel in 1949 – and the role it played in exposing the full spectrum of possibilities that a global understanding of space-time would reveal. Finally, I show how recent studies of so-called supertasks – doing an infinite number of things in a finite amount of time – have shown how global space-time structure determines the ultimate capability of computational devices within them.

Some Historical Background

Physical Impossibilities

Scientific and philosophical consideration of physical impossibilities has a long history (Barrow, 1998). The Aristotelian worldview outlawed the possibility that physical infinities or local physical vacua could be created or observed (Barrow, 2000).